Cloakers have the ability to cloak themselves, making them invisible on the other players' map. These ships can be useful as hit and run pirates or to hide the real strength of your attack fleet. Cloakers can be either a devastating weapon or a useless tech choice, depending on your style of play. They tend to be among the first tech developments chosen by a large number of players, but few players actually use them well.
Most of the common tactics using them can be found on Skai's Cloaker Page. One change to his advice I'd make though, though, is that the best defense against cloakers isn't mines, it is to never allow your opponent to see your planets in the first place.
As you might expect, these are primarily an offensive type of ship. However, they can be useful on defense for those of you who like to spring traps. Most attackers discount the possibility of cloakers as part of a planetary defense fleet, believing them to be a strictly offensive weapon. They therefore plan for a defense fleet having only as many ships as they can see. If you uncloak your ships the turn your enemy moves in, the odds are suddenly different and your fleet survives...possibly to launch a counter-attack.
Another tide-turning cloaker tactic is to use a fleet of them to disrupt an enemy's overbuild. The overbuild will leave all of his ships well below maximum BR, and a few high-BR cloakers can eliminate the entire fleet. Just the threat of such a thing might even prevent the enemy from overbuilding — if they see you have cloakers that might destroy their fleet they might decide to build somewhere else... somewhere less strategic for them.
The other type of cloaker attack on Home Worlds that is common is the longshot, single cloaker attack. Here you hope the enemy isn't paying attention to the map (chatter over the diplomacy screen might help the chances of this) and misses the cloaker. This may be carried out as either part of an ongoing campaign, or it may be utilized as a last ditch effort when you know you'll otherwise lose. A BR-1 cloaker has nuked several inattentive veterans this way. A derivative of this formula is to have a small fleet of cloakers in the enemy's builder/Home World and uncloak only one of them. The enemy then assumes he can build enough normal ships to kill that one cloaker. Set it to nuke and uncloak the rest of the fleet and the odds of a nuke are high.
Personally, I seldom choose cloakers — the other tech developments are generally more important to me. When I do have them I tend to use them mainly as an instrument of harassment. You can often get away with using BR-1 ships for this since the enemy has to assume they are your max BR and defend accordingly...just don't get predictable. Using this strategy you can nuke undefended non-builders and uncloak over builders in order to force the enemy to build. Try to coordinate these appearances with your standard attacks elsewhere —he might not have enough resources to build in both places, thus allowing your fleet to destroy his while your ship recloaks itself and moves on to harass him somewhere else.
Note that the military score your opponent sees on his diplomacy screen does include cloakers, regardless of whether they are cloaked or uncloaked. I've seen statements to the contrary, but at least for version 2.8 they're outright wrong. If he sees a score that is disproportional to the number of ships you have he might be able to guess that you have cloakers. You might want to build decoy BR-1 attack ships to keep with your fleet in order to convince opponents that all of your strength is in your main fleet. Alternatively, you might be able to bluff your way into convincing him that your ships are actually higher in BR than he thought.
One word of caution, if you have a lot of cloakers running around the map your visible fleet will be either underpowered and unable to fight a head-to-head battle, or you will have to build more ships to make up the difference (and thus significantly slow down your tech development.) In my opinion this makes heavy use of cloakers a gamble. If it pays off it will pay off big, but if it doesn't it can cost you a lot. If you tend to be conservative and risk adverse then you might think about using another tech selection.
There are important differences in how cloakers operate on various servers. In older versions of the game, cloakers are built uncloaked, and it requires one turn to cloak the ship before it is actually hidden. The fact that cloakers are built uncloaked means that opponents can see them as they are being built (assuming of course they have explored that planet.) Newer game series have allowed for cloakers to be built pre-cloaked. This has two important repercussions - first, you get an extra turn you don't have to spend cloaking the ship to begin with, and second, you can build them closer to the front instead of having to hide them several jumps back to avoid the enemy's seeing the build.