Most of my experience is in general surgery, so take this info with a grain of salt. People can correct this as needed. In general, the surgical arenas that you can apply to directly out of medical school are as follows:
General Surgery - 5 clinical years of surgical training in GI, hepatobiliary, pancreatic, thoracic, vascular, trauma, surgical oncology, pediatric, plastics, breast, endocrine (thyroid, parathyroid, adrenals, etc.), skin/soft tissue. Most programs (but not all) also include training in cardiac and transplant. Bear in mind that although you get exposure to all these areas during residency, upon completion of general surgery residency most (but not all) will predominantly do more "bread and butter" type cases in the GI, breast, skin/soft tissue arenas. Often, many academic programs will encourage or require an additional 1-2 years of lab time which is often done after the PGY2 or PGY3 years, bringing the total training time to 6 or 7 years.
Orthopaedic Surgery - generally 5 years in length, and the first year is usually a "preliminary" year of general surgery
Otolaryngology (ENT) - also generally 5 years in length, and the first year is usually "preliminary" year of general surgery
Urology - also generally 5 years in length, and the first 1-2 years is general surgery, depending on the program
Neurosurgery - generally 7 years total in length, the first year predominantly general surgery.
Ophthalmology - honestly, don't know much about it.
The other areas of surgery--Plastics, Vascular, Cardiothoracic, Non-Cardiac Thoracic, Endocrine, Pediatric, Trauma/Critical Care, Transplant, Surgical Oncology, Colorectal (I may be forgetting some) all require a fellowship after completion of a full 5-7 year general surgery residency. There are a few exceptions to this rule, namely:
Plastic Surgery - there are some integrated programs which will allow you to match into a plastic surgery residency, which will be about 1.5-3 years of general surgery followed by ~3 years of plastic surgery. If you want more info on this, I would suggest looking at the Plastics message boards.
Vascular Surgery - there are some programs that will also allow you to match into a program similarly structured to the Plastics programs that will train you in Vascular surgery.
Although the surgical fields have become increasingly more competitive, general surgery remains the "least" competitive as compaired to the other sub-specialties listed above. However (and without the intention to start another tired SDN war about who's-more-competitive-than-who) that still means there were more categorical general surgery applicants than there were spots in recent years.
Hope this helps some.