Established in 2161 after the Earth-Romulan War, the United Federation of Planets (UFP for short) is a united government formed by Earth, Vulcan, Andoria and Tellar. This logo was used for the first 120 years of the Federation.
This is actually a canon logo, with a few minor changes by me. Was seen in the last episode of Enterprise, if you want to accept that episode as canon, I don't but here it is.
The later logo as seen in Star Trek 6 and TNG can be seen here:Logo by Majestic aka me.