This may vary from one preschool to another. I know that some of the preschools we've looked into required the child to be fully potty trained before enrollment. Proof of immunization is also important at most preschools.
The answer to this question will depend on whether you are looking at public or private school. The requirements for entry into public pre-k or Head Start are going to vary by state and school district. As an example; in TX the eligibility for public pre-k is 4 years old before Sept 1st and one of the following criteria: low-income (based on national poverty guidelines) or the child speaks a language other than English in the home. Head Start eligibility is based on income. For private schools eligibility requirements will vary by school.