Beatmaking stems from the use of a regular acoustic drum kit in rock, pop, and perhaps especially R&B and funk. Early beats that were made with drum machines or samples were usually meant to re-create beats played on actual drum kits. The most commonly used pieces of the drum kit are just as you list:
- Hi-hat (open and closed)
Many of the most famous sampled beats from funk and R&B that were used in early hip hop use just those three elements (including probably the most sampled beat of all time, the "Funky Drummer" beat). So today, those are the most used and usually the most important elements of a beat.
The other most popular pieces of a typical drum kit, that are commonly used in making beats, are:
- Hi, mid, and low toms
- Crash cymbal(s) (a rock kit normally has a higher crash and a lower crash)
- A ride cymbal
In the late 70s and early 80s, synthesizer manufacturers started to create drum machines, and this really gave birth to beatmaking and the foundation of hip hop and rap. The different drum machines added additional percussion elements to make their devices more appealing to musicians and producers, including:
- Side stick (a way to play the snare drum)
- Hand claps
The different drum machine technologies gave each product its own sound. Many drum machines were expensive and mainly used by producers on high-budget tracks, but two drum machines, both by Roland, were more affordable and became the most popular drum machine for hip hop and rap acts. They were the Roland TR-808 and TR-909.
The TR-808 in particular had a unique sound that was based on analog synthesis, not digital samples (which many other drum machines used). The 808 is legendary for its kick sound, which had a deep and punchy sound that you couldn't really get anywhere else. The term "808" doesn't refer to a kick sound, per se - it could refer to any TR-808 style electronic drum sound - but the most popular 808-style sound is the kick sound from the 808, so you will often come across the concept of an "808 kick", or "let's add a little 808 in there".
Most drum beats serve two purposes in popular music: They add musical interest and they map out the rhythm for the song. One way to make a beat that serves both purposes is to start with a traditional basic map of the rhythm and then make some changes to make it interesting. The basic rhythm map can be made with only closed hi-hat, snare, and kick, and has these elements:
- Hi-hat on every 8th note (8 hits per 4/4 measure)
- Kick on the down beats (1st and 3rd beat in 4/4 time)
- Snare on the up beats (2nd and 4th beat in 4/4 time)
From there, you can vary things in an infinite number of ways. Understanding the basic rhythm map (which no one ever uses as the beat for a song - it's just too boring) will help you hear and understand beats as well as program your own.
Also check out these classic 80s drum machines and their factory or famous beats: