For trombone you'll typically see a number of mute markings, but the style of music changes what you'll frequently need.
In traditional, classical music and other chamber style music you may see the term Con Sord or some similar Italian or even German markings (Con Sordino, Mit Dampfer, etc.). If a mute type is not stipulated, the vast majority of the time you should use a straight mute. In this setting a metal or wooden (not cardboard though) mute is typically desired. Jo-Ral and Dennis Wick are common brands of this type of straight mute.
When it comes to commercial music, mutes are not generally as expensive but it isn't unusual to carry around at least five mutes if not more. However, most of the time, a particular type of mute is indicated.
Plunger or 1/2 (Half) Plunger - This mute is just like it sounds. It is literally a plunger, like what you would use to unclog a toilet, minus the stick. When indicated, what you will do is hold the plunger with the part that would hold the stem between two of your fingers with the bottom part facing the bell. You won't totally close off the bell, but - as the second name indicates - cover it at around a 45 degree angle such that about half of the total direction that the sound can go is blocked.
You will also frequently see symbols "o" and "+". O means "open" and "+" means closed. So if you see a plus sign, it will mean put the mute over the whole bell. Usually this is done for effect such that you will see, for example, one note with a + and the next note with an "O" creating a "wah-wah" style sound.
If you play jazz, this is a necessity as it isn't unusual to at least need this for one song on every gig, if not many more.
Straight Mute - A commercial straight mute is different than the traditional, classical straight mute that I mentioned above. Sometimes. It depends on what the composer or arranger wanted. Usually in commercial settings players aren't too picky about what you play. Some actually prefer the cardboard style mutes such as from the company Stonelined. This will be your cheapest option and also likely 100% fine for any commercial application you have. If you play classical music, you may want to save money and buy a more expensive one as indicated above because most of the time you'll be able to use one of those on commercial music as well. The intent in this style of music is mostly to cause a sharp, buzzy sound and blend will not as often be as important as it is in classical settings.
Straight mutes aren't as common as others in commercial and jazz settings, but you will still see it fairly frequently.
Cup Mute - This is much more common than a straight mute. Again, the stonelined cardboard style mutes are fine. More expensive models exist, such as the Dennis Wick cup convertible - which actually doubles as a straight mute because you can remove the cup for transport. If I was going to suggest one mute, I'd probably suggest that one because it can satisfy both cup and straight needs for commercial, jazz, and many classical settings too depending on the level of the players you are with. However, if you play something like pit music where there may be many quick switches between straight and cup mutes, this may be an unsatisfactory option because it takes around 30 seconds to remove or reinstall the cup portion of the mute.
Bucket Mute - This is another fairly common type of mute, but less so than the three mentioned above. You may not use this on a gig, but over the course of 4-5 gigs you will probably use one - again, depending on the level of the players that you are with. In more informal settings, not everyone may have one in which case you may play in stand, which I will talk about later. There are several types of bucket mute:
In bell - This one sticks into the bell like a straight or cup mute. The primary difference is that the bottom is very deep and usually has some kind of material such as cotton in it to dampen the sound. The goal of a bucket mute is to provide a very soft tone without much edge. A good example of one of these mutes would be a Jo-Ral bucket.
Clamp-on - These types of mutes literally have clamps on the side that attach to the bell directly. They provide a more authentic sound and have much less resistance because they don't insert all the way into the bell, but have drawbacks. First, the mutes themselves have to be exactly the same size as the bell you have. If you have a 7.5" bell, you need a 7.5" mute. And if you want to use it on two horns with different bells, you need two of them. Secondly, they can scratch the lacquer on your bell or even cause damage if you don't put it on right.
Drape-over (softone)- This type of bucket mute is not a traditional bucket mute at all, but rather a type of neoprene that is literally draped over the bell. The sound isn't as authentic as the two above, but they are extremely portable. They can also double as a plunger mute depending on how far over the bell you drape it. They can also triple as a practice mute if you put it over the whole bell. But it doesn't take away much sound. Good for backstage warming up if nothing has started yet, but not so good if someone is sleeping in the same house as you and you need to be super quiet.
The only one of the drape-over that I'm aware of is the "Softone" mut. This is what I use. I primarily use this one because on bass trombone, it isn't super unusual to see notes that require both plunger and the F attachment to be used simultaneously, which is fine if you have three arms... but with the softone, I can just cover it more and it sounds pretty close to the plunger sound.
Hat mute or Derby - Unless you're doing super authentic music, you aren't likely to actually have anyone bring one of these to a gig. You an either substitute a plunger or just play in the stand.
In stand - This is not a mute, but just advice for how to play something that says this. It literally means face your bell into the stand. On some gigs you may have someone not bring or otherwise forget their bucket mute. You can sort of substitute the whole section playing in the stand for that. When you see derby, you can also put your bell in the stand. If the derby requires the "O" and "+" effects, you can use a plunger.
Pixie - This is a mute that fits in the bell very deeply and gives a very raspy sound. Often used with a plunger. You can substitute a cheap trumpet straight mute (stonelined only, the other ones flare too much at the bottom) if you're a tenor player. (DO NOT PUT A TRUMPET STRAIGHT MUTE IN A BASS TROMBONE BELL UNLESS YOU HAVE CUSTOMIZED THE CORKS, IT IS VERY EASY TO GET STUCK). I actually prefer a trumpet straight mute for these applications over the trombone pixie mutes.
You will likely never be asked for this mute combination in an ensemble, but it might come up on combo performances where a very specific style is being emulated. Either mute you choose, you will probably want to buy two sets of corks. Glue the first set on and file them down. You'll notice that it'll start to sound really good and then a little more material will sound really bad. Glue on the second set of corks and file them down to the good spot and then remove the first set. If you use a trumpet stonelined, you can do the whole process for less than $20.
Wah-Wah or Solotone - This is a really niche mute. You will probably only see it in pit gigs from mid 1900s musicals. Unless the director has a specific sound in mind, you can probably substitute this literally without anyone noticing. They can be acquired for around $40 though, so if you really want to, you can pick up one of these eventually but it probably will never pay for itself.
I can't recommend one mute because plungers are inexpensive and just as necessary as a cup so...
...given all of these types of mutes if I had to recommend two I would recommend:
Plunger + Dennis Wick convertible.
If I was to recommend three, I'd recommend:
Plunger + Dennis Wick convertible + Softone
because the'll cover probably 95%-100% of any mute application you'd use in most commercial music.
If I were to recommend four, I'd probably do the same as above but separate the Dennis Wick convertible for a stonelined cup mute and a Jo-Ral or otherwise metal, possibly copper bottomed, straight mute because that would cover more applications and not be much more expensive than the recommendations from above.
A final note: The wood mutes I mentioned earlier are for fairly specialized applications. They alone are rather expensive and not necessary. However, you may much prefer the sound of wood over metal or cardboard mutes. If you have a section that is willing to all shell out the money for the wooden mutes (such as from the company Facet) you may be very pleased with the results. However, you'd probably also want to have a set of the ones above because they are much more common and will blend better with other gear that the typical player will have.