"Bumper" I've only heard used to refer to audio clips used to segue between sections of a show, or in/out of commercial breaks. I was surprised to see those other answers mentioning they've heard "stinger" used for that purpose, I haven't.
These wikipedia pages have more info for the usage of "bumper" that I'm familiar with:
Bumper music, or a bump, is a term used in the radio broadcasting industry to refer to short clips of signature or theme music used to buffer transitions between programming elements. It is also a term for music played at music venues such as concerts before show time, to fill the air, with a musical atmosphere. Bumper music is commonly employed when a syndicated program takes a break for local station identification or "goes to a radio commercial." More often than not it is called a "bump" in today's radio; NPR also uses the term "button". It is also referred to as "rejoiner music" when the bumper music marks the end of a local break on a radio network.
In broadcasting, a commercial bumper, ident bumper, or break-bumper (often shortened to bump) is a brief announcement, usually two to fifteen seconds in length that can contain a voice over, placed between a pause in the program and its commercial break, and vice versa. The host, the program announcer, or a continuity announcer states the title (if any) of the presentation, the name of the program, and the broadcast or cable network, though not necessarily in that order. On children's television networks, they are sometimes called external eyecatches due to the resemblance of internal eyecatches in anime and there is usually no voice over, but some bumpers do feature one. Bumper music, often a recurring signature or theme music segment, is nearly always featured. Bumpers can vary from simple text to short films.
Personally, I've never heard "sounder" used for any of these meanings. I interpret "logo" as a visual representation of a brand (not audio). And "jingles" are the catchy melodies/songs that companies use in marketing, ranging in length/complexity anywhere from the short-n-sweet McDonald's melody+lyrics "ba-da-ba-ba-ba.. I'm lovin it," to the full "Meow Mix" cat food commercial song.