These are the ultimate dreadnoughts of a fleet. They have the ability to not only nuke a planet, but completely annihilate it — reducing the value of all resources to zero and making the planet incapable of being recolonized. Note also that doomsdays can obliterate not only an enemy planet, but also any uncolonized one as well.
Just the presence of a doomsday ship can change the enemy's entire strategy. As one long-time player put it, they change the equation from "I'm too busy to defend it. I'll just recolonize and accept the loss" to "It's a total loss if I don't defend it!" This, then, may cause them to shift resources away from another front and allow your attack there to succeed.
In spite of this vast power, doomsday ships are among the least used of all the ships in Stellar Crisis. This is largely due to their expense — in most cases the value you get simply does not justify the cost. There is also the probability that you would want to colonize a planet after nuking it, which the use of doomsday ships makes impossible.
There are two legitimate uses for the doomsday ships. One would be to annihilate a planet that you know you will not be able to hold. It is common to nuke and colonize a planet, but then not have enough ships close-by to defend it. In this case you might want to annihilate the planet and make sure that the enemy cannot renuke and recolonize it. This is especially the case if he is using a stargate that could send in a new fleet the next turn. Annihilating the planet could prevent his ships from gating directly to he front, as well as preventing him from regaining *any* of its lost resources.
The other use for doomsday ships is risky, though I have done it and it can work. Select doomsday ships as your first tech selection, and build one along with your first fleet. If you can quickly explore and annihilate one or more of the enemy's planets you will have effectively denied him those resources for the entire game. If it was a large-resource planet then can be the deciding factor in the game. The downside to this tactic is the cost of the doomsday ship and the lack of other tech selections if the attack fails. Building and maintaining the ship will severely hamper your tech development at such an early stage of the game, and there is also the chance that the enemy could kill your ship before it has the chance to destroy any planet.
In general, doomsdays are a fun toy to play with once you have high BR and the game has been pretty much decided. They probably aren't worth a tech selection until you already have all the ship types that you really need. Perhaps the most important value of the ship is using it as a newbie detector, since newbies tend to be swayed by the description of what the ship can do and build lots of them.