Birth records can be a genealogist's nightmare. We've all run into a situation where the birth record is key to proving our case and we have no idea how to locate it. Many states did not require registrations until the mid-20th century. Even if the birth was registered, you still need to figure out how to access it. Some records are only available at the state level, others only locally. It's always complicated.
In Connecticut, the process is actually fairly simple. Births were recorded statewide after mid-1897 but do exist before that date in certain towns. Connecticut is a closed record state for birth records. To access a record dated after 1911, you need to be a direct descendant or a member of an incorporated genealogical society. Visit http://www.americanancestors.org/connecticut-vital-records/ for more information. Once you have a date of birth and an understanding of your ability to access the records, it's time to actually request the record. The fastest way to get a record is to approach the Town Clerk for the town in which the event occurred. You will have to complete their paperwork, likely provide identification, and pay a fee. If you are unsure of the date or location of the event, apply for a copy from the state office of vital records. Their application process is described on their website: http://www.ct.gov/dph/cwp/view.asp?a=3132&q=388130. Beware, they have a long turn around time.
Good luck with your search!