When you finish medical school, you earn the ability to be essentially on-the-job trained in the specialty of your choice(that's what a residency is). A residency pays you(albeit not much compared to attending salary) to get good at your specialty.
Because there are only so many hospitals, and an even smaller amount of them that could sustain a residency seat, there are limits to how many seats there are per specialty.
The AOA sponsors residency seats just for DO students that are available after you take the COMLEX. These tend to be focused on the primary care specialties(family, peds, OB/GYN, etc.), but there are slots available for just about every specialty.
There are allopathic residencies too- and lots of them. After all, there are a lot of allopathic students and there are residencies that cater to just about every specialty here too. What a DO student can do if they want to go to an allopathic residency is to take the USMLE and try to match into that residency.
A DO will have an easier time with getting into a primary care specialty simply because that's what a majority of the residencies are intended to train someone in. Although, it is not impossible for a DO to get into a more competitive specialty. You have to take the USMLE and kill it, though.
At least that is my understanding. ^^