I've heard terms like "hook", "bridge", and "chorus", but I'm fuzzy on exactly what they mean and if there are other parts.
The most common parts of a song are:
Verse - Usually longer, more complex, more narrative, less melodic, often in a minor key or a darker mood, featuring more challenging lyrical content, usually sung by a soloist. There are usually two or more verses, and they usually have different lyrics but the same melody. (Some, or all of these characteristics can be missing).
Chorus - So called, because it's traditionally the part where everyone joins in and sings together. Usually shorter, simpler, non-narrative, in a major key or happier mood, less challenging and more reassuring. Usually is identical in lyric and melody each time it is repeated (sometimes it has minor variations). A brief, one-line chorus, that alternates with material from the verse, is called a refrain (although the two terms are often used interchangeably). A chorus sometimes has its own intro ("pre-chorus") and/or outro ("post-chorus") section. In a song without a hook, the chorus is the part people remember.
A lot of songs in a lot of genres, from a lot of eras, all over the world, have verses and chorus. It's a very natural structure if you want to combine a soloist with a group. Most older songs start with one or more verses, and delay the chorus. A classic song in this format is Kenny Rogers' The Gambler. Newer songs often put the (audience-pleasing) chorus first, because the ability to skip songs means that songs that get off to a slow start may never find their listeners.
Other parts of songs vary by genre. Here are some relatively common ones.
Bridge - Very common in classic American pop music. This is a section added to add interest to a song. It is usually a contrast to both the verse and the chorus, and often uses comparatively unusual chords. The lyrical content can provide a different perspective on the song. It is so-called because it is a different connection (or "bridge") between the verse and the chorus. Examples: Still Crazy - Paul Simon, Long Time Gone - Dixie Chicks. It is often just instrumental, as in the horn bridge that Aretha Franklin's producers added to Otis Redding's Respect. In modern pop and R&B, the bridge often takes the form of a rapped "verse." Example: Waterfalls - TLC. Some people call the bridge "the middle 8" which is a technical description of where a bridge is usually placed, and how long it usually lasts.
Hook - Characteristic of modern pop, and a tribute to short attention spans, the hook is a brief, memorable, often quirky "earworm," an addictive bit of melody or lyric designed to keep a song in people's heads (whether they want it there or not). Rap songs often have hooks in the place of choruses. An example of a hook is the "ella, ella, eh, eh" section in Rihanna's Umbrella. The hook has become so important to current pop that the new rule is "always start with the hook." It's still the "hook" however, even if it doesn't come at the start. The name refers to the role, not the placement, which means the category can overlap with some of the older descriptors, most typically chorus or bridge. (Note: Some songs have more than one hook, and some are arguably all hook.)
Intro - Common in old Tin Pan Alley, classic Broadway and Great American Standards songs, the intro is a long, mostly non-melodic, narrative intro to a song (it was sometimes called "the verse," but there would only ever be one). Often omitted in later performances. Examples: Somewhere Over the Rainbow, Blue Skies.
Call and Response - Common in traditional folk music, old school gospel, and modern K-Pop. The call-and-response is kind of a hooky mix of verse and chorus, where the lead singer sings a line, and then everyone else sings the answering line together. Example: Snuper - Tulips
You can mix and match any of these pieces, but typically most songs in a single genre will share the same basic structure. A common American pop structure is "Verse Chorus Verse Bridge Chorus", but there are infinite variations.