An undersea home has certain pros and cons that set it apart from any other bases or homes. Firstly, an underwater home is immune to any explosions, is difficult to access by Mobs, and can provide excellent visual range and protection. However, an undersea fort can be somewhat time-consuming to build, enter, and exit. Follow these directions and you will have an excellent undersea fortress. As a general rule, building underwater is greatly enhanced with a helmet enchanted with respiration and/or aqua affinity.
A detailed video guide about underwater bases can be found here.
There are several basic techniques that are used and combined for underwater work. The most immediate issue is not drowning while you work, followed by clearing the water out of your new base. If your base is deep, getting there and back may also require special measures.
 Breathing underwater
Armor with the Respiration enchantment can extend your breathing time underwater, but even so, for extended work, you will need to create an air bubble that you can retreat to to refill your breath meter. There are many partial blocks which don't occupy all (or very much) of their space, but which can force water out of their entire block cell. Fences are probably the cheapest of the lot; if you stack two fences on the sea/lake bottom, you can stand next to them to breathe freely. (You need two so one will be at head height, but if you like you can then mine the bottom one.) Cobblestone walls work exactly the same way, while glass panes and iron bars occupy a bit more of their bubble.
Several other blocks can also be used, but unlike fence-type blocks, these will break and drop if the block they were attached to goes away. A door is slightly awkward (and unstackable), but you only have to place one thing to get your bubble. However, if you have a vertical surface or block available, you can easily put a sign, ladder, trapdoor, or any of the above blocks against it at head height. You could stack two signs on the floor like the fences, but stacking signs is a bit picky, which is the last thing you want when your air is running out.
All of the above can also be used to block water out of entrances/exits which are generally a good place to start building. To make an air/water doorway from fences, glass panes or iron bars: Pick a corner of your underwater home, and place the fences or panes as through you were making a 2-block high corner window, but skip the actual corner. The two diagonal blocks will keep the water out, but you can easily walk between them.
As of version 1.7, advanced players may have access to Potions of Water Breathing, which will allow them to stay underwater for long times. However, it's still wise to have some bubbles handy just in case you run over time, and/or run out of potions.
In relatively shallow waters, you may be able to just swim down and place a "bubble block" as above. If your target spot is deep enough that getting there is risky, you can make a pillar of fences, glass, etc. from the bottom to shorten the trip. (In SMP, you may need to have your pillar stop well short of the surface, for concealment. If this is not an issue, the pillar can go all the way to the surface.) If you want to go deep enough that you just can't swim down there, your best bet is probably to drop gravel or sand to make a starter pillar. Then you can run ladders down it from the surface, for initial access. However, you don't want to depend on this setup for long, because if any of the blocks are mined or even blasted, all the ladders above the break will drop. To make the pillar permanent, dig down, replacing the sand/gravel with bubble blocks as you go. You can even replace it with cobblestone, dirt, or, whatever and then stick ladders on that. Alternatively, you can go down to the bottom, and make a new pillar somewhere else, or arrange for other means of access. Lily pads can help with placing the gravel, or let you run a fence or glass pillar from the surface downward. (Put one block on the pad, then use the side of that to place other blocks.)
 Underwater Access
The simplest sort of underwater base is made by simply placing a door against an underwater mound (you need at least a 3×3 space of dirt or stone to start with, and a floor block to put the door on). Once you open the door and step into it, you can simply continue to dig out a lair more-or-less if you were aboveground. Once you have an entrance passage, you can place ladders or signs (or another door) inside the original door for a flush entrance, and remove the starter door. If you like, you can then turn around and clear out some of the water to continue your base outward.
For an even more inconspicuous entrance, you can start digging a 1×1 shaft down until you hit stone, or a little more, placing ladders every other space as you go. Watch out for caves-- if you break into one, you can either switch to securing the cave and use that for your base, or go back up, seal the hole, and dig somewhere else. Once you hit stone, mine the first two blocks down, then come back up and start digging horizontally. (This will divert water coming down the hole away from your horizontal passage.) Continue to hollow out your base "until it be enough".
 Clearing Out Water
If you instead want your base to be truly "underwater", you will need to clear water out of a fairly large volume. The only real way to do this is to fill the space with blocks, and then remove most of them. So, construct rooms with walls and ceiling, but fill the inside with your chosen filler block. You can use an easy-to-break block such as dirt, wool, or even (if you already have a floor) sand or gravel. Alternatively, you can use a burnable block as filler -- wood planks or wool will work, but leaf blocks burn fastest. When you've got your space filled, mine out one of the blocks and use a flint and steel to set a fire in the space, wait for that to burn out, then repeat as needed for any unburned sections. Place jack-o-lanterns or glowstone to light the new space, to prevent spawning. (Torches would be washed away if water does ever come in, say while you're remodeling.)
For walls and ceiling, almost anything will do, but glass, fences, or iron bars give a nice view, and add options for entrances from the water (see below).
 Drop-in construction
This method is particularly good for building in very deep areas of water as it requires less time spent underwater.
Find a fairly deep patch (depending on how big you want your home to be) of water with a relatively flat floor. Then, on or above the water surface, build a platform out of any non-gravity block. (It's best to use blocks you can remove easily -- dirt, wool, leaves, etc.) Build the shape of your desired building on the platform out of gravel or sand, as a solid block or "sand castle", but without the exterior walls. When you are happy with your work, dig or burn out the platform underneath (beware of falling sand!) Then all you have to do underwater is cover the fallen mold with walls and ceiling, and hollow it out. A tip for hollowing: Bring down a slab or stair of any kind. Use it to dig out the bottom block of sand and immediately place it on the bottom of the space beneath. This may take a couple of tries, but when you get it in, the rest of the sand will fall on it and break.
 SMP Hiding your Base
In Faction bases do Not claim land as anyone going over by boat or under by pick will notice the change from wild area to faction area. Make it as deep as possible if possible on the bottom of a deep ocean with a central hub (where f home is set) made of blue glass or a gravel "hill" with a blue glass tower. make sure half slabs are used over any kind of light near windows or exits. Only where faction members need it badly. Farms should of course be lit but make sure there is no hint of light coming out anywhere. Keep buildings seperate, small, and blending in irregular shapes. Blue glass is the only glass that should be used as it blends in the best with water. The best thing about bases underwater is you can if carefully planned mold entire biuldings. also keep player detection systems stationed in a ring around 40 blocks away from where members should be and buy from server shops cow spawn eggs to supply the cow based player detector. But the best test is going up to the surface and drink a night vision potion after looking at where your base is located to look for any obvious signs of a base then do it under the effects of the potion. If you do see those signs go back down and troubleshoot. (just be aware of the x y and z coordinates as the the author of this paragraph is still trying to refind his underwater base with his coordinates)
- Sugar cane blocks water, but not movement. You can use them as air to water doors. However, placement is restricted: Just like on dry land, the bottom block must be placed on sand or dirt, with a water block next to, and one level below, the placed cane. (In version 1.7, underwater gravel becomes common, and sugar cane can't be placed on gravel.) Also, sugar cane is fragile, and if any cane is broken, all the canes above it will break too.
- Ice may form over the surface of your chosen water, generally in colder biomes. Make sure to arrange a way to get past it safely.
- Buckets can be used to regain your breath. If you need air, you can click an empty bucket anywhere (no target needed) to refill your air meter. However, that will fill the bucket. To empty the bucket so you can do that again, you need to click it on a block.
- Torches can be used almost like signs and such for temporary air pockets -- they will drop almost immediately, but if you're in their space, they will replenish your air meter.
- If building an underwater tunnel to access your base, remember that still water will allow you to pass in both directions at the same rate, but flowing water works better for one-way paths.
- You should always make a second door and chamber just in case one door breaks and your house is flooded.