Strategic concepts, tactical patterns, openings, and other resources
Note: this article was posted on reddit first.
According to the wiki:
Crazyhouse (also known as Drop chess, Mad chess, Reinforcement chess, Turnabout chess and Schizo-chess) is a chess variant similar to bughouse chess, but with only two players. It effectively incorporates a rule from the game shogi, in which a player can introduce a captured piece back to the chessboard as his own.
The fact that pieces can be re-introduced into play makes this an extremely tactical game with a huge emphasis placed on initiative and solid structure. You have to be able to calculate not only the normal chess variations from any given position, but also the resulting implications that could arise with each exchanged piece; and, even more than in chess, you need to actively prevent weaknesses in your own structure.
Pawns and knights increase slightly and the queen decreases slightly in relative importance, and there is no endgame to speak of, since all the pieces can be placed back on the board. Otherwise, crazyhouse takes all the normal chess themes and heightens them by magnitudes. An extra tempo or two can lead to a crushing attack, and even small weaknesses, when exploited correctly, can become glaring. Precision is almost always required, especially when defending, as every position is a sharp one.
For those reasons, I'm willing to propose that becoming a proficient crazyhouse player can only help your chess vision: you'll see tactics more naturally, calculate more efficiently, and identify weaknesses more quickly. A lot of ideas can be carried over. But even if that's not the case, crazyhouse is an extremely fun game that can stand on its own merits! So let's see what it's all about.

I should start by conceding that (as of yet) there's little or no "theory" as there is in chess. It's all pretty much touch-and-go, and the bulk of the action will take place in an often-explosive middlegame. But there are definitely some rules of thumb to live by.
As white, you have an extra tempo which you should use to at best start an attack and at worst gain a positional advantage. As black, your goal is first to neutralize white's initiative and then equalize by imbalancing the position or counterpunching.
The standard rules apply: develop your pieces, control the center, and get your king to safety. But there are some opening ideas uniquely emphasized in Crazyhouse:
The middlegame is almost always where the game is won or lost. This is where you should look to start exchanging pieces, breaking with your pawns, building up pressure on vulnerable squares, or cracking open your opponent's defenses with careful pawn drops or piece sacs. The midgame is rapidly achieved in crazyhouse, so be ready to join battle straight away! I'll cover strategic midgame motifs below in the 'Strategy Pointers' section.
Lots of exchanges will take place in the middle game. When considering the tactical implications of exchanges, remember that, though they move the same, pieces accomplish slightly different things in crazyhouse than in chess, and that should be accounted for in your mental calculations. As a general rule of thumb, I like to think of piece values thusly:
lol what endgame

In this example, without the g-pawn, the dark squares, particularly g7, are weak. So white places a pawn on h6 to control that square, and will then look to exchange for more pieces, especially diagonals, to drop on black's dark kingside squares. Notice that a situation like this makes a queen trade highly desirable for white and losing for black, giving white even more control over the game.
Here's my Crazyhouse opening repertoire, listed by the frequency with which I play it. I'm not going to claim it's anywhere near perfected, or even that it's what you should play. But I've had success with it. Obviously the move orders will change quite a bit depending on what your opponent does and you'll have to be flexible, but I'm going to list the common move order for each just so you can get a feel for the general set-up. I can't go through each variation because it would take forever, but if there's a certain line you're interested in let me know.
Basic setup: 1. d4 ... 2. g3 ... 3. Bg2 ... 4. h3 ... 5. Nf3 ... 6. Bg5 ... 7. 0-0 ... 8. Nbd2
Illustrative Game: lichess.org/THYYswro
Themes: Solidify and fortify your king before you attack. h3 is an important move so your opponent can't place a pawn there attacking your fianchettoed bishop. You start out a bit passive but extremely solid. Black will have a hard time breaking through while you use the time he's trying to drum something up to form an attack of your own on his likely more exposed kingside. Trade pieces in the center and then drop on the kingside. Often p@e5 to challenge the center or a pinned piece or p@h6 to pry open the king. You can allow the f3 knight to be captured, whereafter you'll often recapture with the e-pawn to build a nice box around your king. If they sac on h3, capture and then simply replace the pieces right back where they were. If no tension develops after the first 8 moves and there is no obvious attacking idea, bring a rook to c1 and break with c4 (or just play c4 immediately) to trade pawns and open things up.
Basic setup: 1. d4 ... 2. Nc3 ... 3. Bg5 ... 4. Nf3 ... 5. e3 (or e4, if allowed) ... 6. Be2/d3 ... 7. 0-0
Illustrative Game: lichess.org/LTdPNcHW
Themes: Focus on the pin on the h4-d8 diagonal. Sometimes exchange for a second bishop which you can place behind the first on h4. Pile up on the pin. Option of trading on f6 for a pocket knight or on e7 for a pocket bishop. Try to advance e4-e5 if given the chance and claim space. If the black bishop goes to f5, challenge it with Bd3 and re-capture with the c pawn for a strong center; if it goes to g4, block the pin with Be2. If given the chance, exchange pawns in the center and place p@h6. In completely passive or dubious black systems where they're generating little or no counterplay it can be explosive to play e4, d4, and then after the minor pieces are deployed f4!? followed by 0-0 and f5.
Basic setup: 1. e4 ... 2. Nf3 ... 3. Bc4/b5/e2 ... 4. Bg5/f4 ... 5. Nc3 or Nbd2 followed by eventual c3 ... 6. 0-0
Illustrative Game: lichess.org/UA6ayvXz
Themes: Develop, control the center, castle, and then immediately attack. I don't play e4 so much, but you can get a good feel for it by reviewing some of FICS's top-rated player's--tantheman's--white games. Find them here.
Basic setup: 1. b3 ... 2. Bb2 ... 3. e3 ... 4. d3 ... 5. Nf3 ... 6. Be2, stuff like that.
Illustrative Game: lichess.org/dkZPL3rf
Themes: Throw your opponent off. Build a solid structure and ask your opponent how he's going to break through. Castle queenside occasionally. Break with c4 or throw your kingside pawns at him.
This affectionately self-titled opening is one I developed through many games of trial and error on FICS, and it's my go-to weapon as black. It looks dubious at first glance but it's held up even at very high levels of play. Its themes can actually carry over into several other opening variations in which you allow your f6 knight to be captured.
Basic setup: 1. e4 Nf6 2. e5 Nc6!? 3. exf6 gxf6! 4. ... d5 (can transpose with 2. ... d5 instead of Nc6) 5. ... Rg8 6. ...Bg4 7. ...e6

Illustrative Game: lichess.org/W4lzqvxx
Themes: Firstly, you build a strong center around your king. The pawn on f6 prevents a knight from coming to the key g5 and e5 squares where it could attack f7. Then your rook is coming to the g file where it's going to dictate play on the kingside and constantly pressure g2. You'll often have a pin on the h5-d1 diagonal where you can win your piece back by placing p@e4; otherwise, that pawn will often be placed p@h3. In return for the piece you get a pawn that you'll use to severely weaken white's position and tremendous pressure on his light square complex in the upper left quadrant of the board (near his king). Capturing the knight on f3 and then dropping N@h4 where it attacks both f3 and g2 is a common motif. Your king will always remain in the center where it's well-guarded. Be careful not to let your h-pawn fall or too much pressure to be dropped around your rook or your king will be come exposed quickly. All in all, you're offering to temporarily go down in material for an open, exciting game where you have immediate counterpunching chances that white will be hard-pressed to deal with without precise play.
A second variation deviates on move 3 with 3. ... exf6 instead of gxf6 and continues 4. ... d5, whereafter you develop normally. This is perfectly viable. You offer a knight for a pawn to achieve open lines and accelerated development.

Basic setup: 1. ...g6 2. ... Bg7 3. ...Nf6 4. ...h6 5. ...d5 6. ...Bg4 7. ... 0-0 8. ...Nbd7 or Nc6
Illustrative Game: lichess.org/Ix9npN7B
Themes: This is going to operate much like the modified Catalan I described above, except for black. You build a strong little box around your king (see image above), whereafter you break with d5. White will often push or place a pawn on e5, and you can either move the knight or use the tempo to start an attack or do something else constructive with the idea of allowing the capture on f6 so you can recapture with the e-pawn and build your box. Exchange in the center and drop on white's kingside. King safety is paramount; keep your box fortress stable with pawn drops as needed (p@e7, p@g5, etc.) and then counterpunch. When attacking pawns can often be placed p@e4 to challenge a pinned piece and gain central influence or p@h3 to open his king.
Basic setup: 1. ... Nf6 2. ... Nc6 3. ... d6 / 3. ...d5 / 3. ... e5. I prefer 3. ... d6 4. ... Bg4 5. ...e6 6. ... Be7 7. ... 0-0
Illustrative Game: lichess.org/Hjrcklag
Themes: Get your pieces out quickly and build a contained, modest setup staring down white's center without creating tension (d6 e6) or create immediate imbalances by breaking in the center with a quick d5/e5. Often castle kingside, sometimes leave it in the center.
This should be the go-to opening for new players. It's solid, simple, and avoids most opening traps.
Basic setup: 1. e4 e6 2. d4 d5
Illustrative Game: lichess.org/o36HZqS9
Themes: In the exchange varation (3. exd5 exd5), you're happy because your light square bishop gets active and you have open lines. In the advance variation (3. e5), you will not break with c5 or f6 as you might like to in standard chess. Instead you can continue by getting your knight to f5 (often via e7) and your bishop to e7. Because the f6 square is temporarily weak when your knight comes to e7, I actually prefer slightly varying the move order with 2. Ne7 before breaking with d5 so that your knight can immediately move after you break, uncovering the queen's defense of f6. You can also bring your knight to f6 first and move it to e4 after you break with d5 and he advances (or accept the capture on f6 with gxf6 and enter into Crosky Gambit territory!). The main advantage of this opening is that it discourages the Bishop from coming to c4 and peering down on f7.
Basic setup: 1. e4 d5 2. exd5 e6 3. dxe6 Bxe6 4. ... Nf6 5. ... Be7 or 1. d4 e5 2. dxe5 d6 3. exd6 Bxd6 4. ... Nf6 5. ... Bg4
Illustrative Game: lichess.org/YydlL3bB
Themes: Gambit a central pawn for compensation by way of accelerated development and open lines. Best played against weaker opponents who can be confidently outplayed, but can be a fun surprise weapon against strong players too if used sparingly.
Beginners may be best served by playing conventional e4/d4 systems as white and the French as black. Avoid pawn moves which weaken key squares; fortify weak squares around your king. Maintain the initiative and attack. Sac when it draws the king out and you have a follow-up. Emphasize king safety over material gain. Calculate what your opponent (and you) can do with exchanged pieces before entering into tactical complications. Go crazy!