As a D.O., you can enter either an osteopathic AOA accredited residency or an allopathic ACGME accredited residency. Currently, some programs are also offering dual accredited residencies.
In an osteopathic residency, the first year is usually a "traditional rotating internship" year. This year is incorporated into osteopathic GME because it is believed that it makes for a physician with a broader scope of knowledge and someone with more experience than a traditional PGY1 (providing a greater level of confidence to handle cases in the second year of residency). If you enter an osteopathic residency, you will not be wasting a year by doing the internship year. If an ACGME residency of the same specialty is four years, the analogous AOA residency will be a 1+3 type program (internship + specialty training).
This works in cases where the ACGME residency requires the PGY1 year to be an internship year; ie. Internal Medicine and Anesthesiology to name a few. Some residencies, like those of radiology, which are extremely competitive to enter, do not incorporate an internship year at all.
When you enter the ACGME match (optional), you can either match for a PGY1 position or an advanced PGY2 position. If you fail to acquire an ACGME PGY1 spot, you will probably be forced to do an osteopathic internship year (assuming you can still find one, since the osteopathic match occurs 1 month before the allopathic match). In some cases however, you may have matched for a PGY2 spot, and you would immediately transfer into that year following completion of your internship. This works for residencies that will count your internship year as a traditional PGY1 year (or so I believe).
If you fail to match either ACGME PGY1 or PGY2, you can do an osteopathic internship year and attempt the match again for the following year. You may either match to a PGY2 position or you may be required to complete your PGY1 year all over again (in the latter case, you would be taking an extra year as opposed to the usual route).
There are five states that require an osteopathic physician to complete an osteopathic internship year (a traditional rotating internship) in order to acquire licensure and be able to practice lawfully. These are MI, FL, OK, WV, and PA. Under Resolution 42, you can petition your ACGME PGY1 year to be treated as an osteopathic internship year. From what I've read on here, it's more paperwork, but it's hardly ever rejected.
A fellowship is something you can choose to do after residency (ie. a cardiology fellowship after an internal medicine residency). The internship year, as stated in a previous post, is classically referred to as just the first year of residency.
Please feel free to correct me if I'm wrong. These things definitely look confusing at first, but I wouldn't really consider them to be negatives. Remember, D.O.'s can enter into either AOA or ACGME residencies. We have a lot of options. These misconceptions of the osteopathic internship year being an extra year only come into play if you consider transferring to an ACGME residency later on. In most cases, the place you transfer to will consider your OGME1 year equivalent to their PGY1 year.