So, IMHO, you've got a few main aspects to a game's combos: execution, length, paths, variety, and flexibility.
Execution is what most people think of first when they think of combos. Good execution is a requirement to use combos, with good varying by game. Shorter combos are easier to execute, longer combos are harder. Let's go through some games. (I'll also start to tackle length here.)
- If you want a game without combos, play Divekick.
- Super Street Fighter 2 Turbo and Fantasy Strike are games that focus heavily on short combos that are based around jump-in > normal x special (x being cancel). You have to time your jump-in button correctly to leave your opponent in hitstun long enough for you to throw out and hit with your grounded normal, and then execute your special move before your cancel window is up. So, with Ryu, you might do jHK (jumping heavy kick) cMK (crouching medium kick) fireball (down>downforward>forward+heavy punch motion). This is short, but feels alright. The timing is satisfying because it rewards mindfulness, and the special cancel is satisfying because you get to slide your hand along that quarter circle from that cMK, and also confirm off of your jHK to know whether to stop at the safe cMK or keep going into the fireball.
- A game like Ultimate Marvel vs. Capcom 3, on the other hand, will have long combos that are a bit more forgiving on timing, but require you to remember them during the heat of a match. UMvC3 tends to have some more repetitive parts during combos, which some people really enjoy.
- Marvel vs. Capcom 2 or the original MvC, on the other hand, mostly consist of shorter combos of ground string > launcher > air string > super cancel. Some people really like how that feels. Once you get in, the base structure of combos for characters is going to be the same, so you can use a lot of the large roster without sucking.
- Skullgirls is big on combos. If you get set up properly, you can go for a while, with varying levels of repetition depending on what combo you're going with or how you decide to use it. This long execution time opens up the possibility of dropped combos, and brings the other player's "execution" into more play. I'll probably cover that at the end with general notes on how combos affect the rest of the game. Anyhow, Skullgirls has some long combos that can be really satisfying if you like how your stick or controller buttons feel and how the chains roll. A big base of Skullgirls combos is the "magic series": you can cancel LP>LK>MP>MK>HP>HK, with any of those being crouching, and you can run this series in the air. You can also skip whatever you want (your cancel may not hit, but it will cancel). You can't run this backwards, but some games do feature running reverse magic series chains as parts of their gameplay. (Melty Blood has this on certain moons, and I think FighterZ has it during sparkling blast.)
- I don't know much about Guilty Gear combos, but I do know they force you out of loops pretty quickly due to how air juggles end up increasing the gravity of your opponent, making them drop out. The way combos flow and evolve here can feel really good.
- Street Fighter 4 has a lot of instances of "links": combos done by taking advantage of timing instead of cancels. A 2-frame link may show up when you throw out a LP, your opponent goes into stun, and you have 2 frames to properly get a MP out before the combo fails. This is obviously hard to do. Focus Attack Dash Cancels are another big part of both SF4 combos and neutral. You can cancel an EX special move into a focus attack, which allows you to then follow through or dash out. You can make an EX DP (which costs a bar of meter) safe on block when you'd normally be open to an easy punish by cancelling into focus attack and backdashing before they can get out of blockstun and punish you. In a combo, you could use EX DP to FADC to lanuch them up for further juggles. FADC with other specials also leads into a lot of Ultras.
Length is closely tied to execution. Some people like shorter combos, and some people like longer combos. Short combos are obviously easier to remember, but long combos aren't as bad as beginners normally think. As you get a handle on your character and fighting games in general, you'll start to get a feel for where moves really are, and where they might mesh. Combine that with some lab work and combos like "cLK sMP sMK cHP / jLK jMK jHK (Steer Down) / s[MP] sMK cHP / jLP jMK jHK (Steer Down) / sLK s[MP] / sLP sLP s[MP] sMK sHP x H Danger xx Cannon" (from http://skullheart.com/index.php?threads/robo-fortune-combo-thread.6797/ ) start to make a lot more sense as you can visualize them in your head and they mesh with how you think moves should flow. Longer combos tend to open up more avenues for creativity and expression, which we'll get to later. Shorter combos normally lead to more chances to play neutral, and longer combos normally push rounds to a couple of key neutrals punctuating those longer combos. Momentum becomes a big thing when you don't have as many neutrals. This can make footsies and spacing even more important (which is where stuff like assists generally fall in team games, as they can cover abnormal parts of the screen that your character normally couldn't touch).
Combo paths add a lot to the richness of a system and help prevent a game from feeling stale. If you always go into one combo, that's boring. (See variety.) But if you can get the same starter and go into a couple of different combos that are all good, that's really cool because it works your decision-making and, if they're different lengths, keeps the opponent on their toes, as well as giving them a vew into what you're going for if they know your character. Normally, you'll have a subset of the path types: decent, reliable damage, high damange but high execution risk, more meter but less damage, more screen carry but less damage or meter, corner options if you've got your opponent over there for higher damage or meter. possibly with less risk than other paths, or even something that trades them being in the corner for something else. Longer combos open up more options here through resets. Normally in games with long combos, damage scaling is very important. It's what makes the first part of that combo do 60% and the remaining 2/3rds do 40% of the total damage. Through resets, you can stop the combo, reset scaling, and then pick up a new one before you really get back to neutral, if you play it right. Long combos offer a lot of opportunities to drop and reset, with some good and some bad. The opponent has to keep on their toes to block, jump, or tech out if you go for a reset, because that can get them back into neutral and give them a chance to get rolling without the full damage they could have taken if you didn't go for the reset. Games like Skullgirls go further by having bursts that the opponent can pull out in really long combos to escape. You can bait these out, though, if you know what you're doing, and get an easier reset. Also, games with longer combos tend to have them because they give the player a lot of choice.
Variety was already mostly covered, but basically, watching the same combo over and over again is boring. You want to see different stuff depending on the situtation to showcase player skill and knowledge. Good variety ties into something important to combos—hit confirms. Sometimes, you have to reach on a combo starter, and if you whiff or get blocked and try to run the rest of the combo, you'll make yourself unsafe and get punished as opposed as just returning to neutral or being able to continue applying pressure. Confirming those hits into combos shows that you had the presence of mind to know where you needed that situation to go and how to get there, and that you could hold onto that confirm while in a dynamic situation that might in a moment call for a different confirm if your current attempt was declined and you had to keep trying differently. Some confirms are really tight, and seeing those be pulled off, even if the combo isn't the greatest, can be really impressive and hype-generating. This ties in with movement and neutral tools in general, as choosing the right approaches and being able to make those stretch confirms may require utilizing some uncommon mechanics.
Finally, flexibility (or expressivity, or openness). Some games just don't let you get that far combo wise. Mid and low tier MvC2, while being satisfying, doesn't let you reach that much combo-wise. Neither does SSF2T. A game that does, that I haven't brought up yet, is the Street Fighter Alpha series (aka Street Fighter Zero). Alpha 2 and 3 had a custom combo system that really let you abuse a ton of stuff if you had the meter, and you can really see how a player likes to do things depending on how they use the system. Same thing with King of Fighers, or Skullgirls, or Guilty Gear, or a lot of anime fighters, or Smash/Rivals (due to the flexibility of the movement-based mechanics and bevy of options the opponent has, but then again they're not traditional FGC and don't have that many true combos). You're able to kinda go freeform to an extent due to how many paths are available and how many of them aren't bad. Even while you're doing something you've maybe labbed out for hours, you're doing a combo that's uniquely yours, and fits your kit and way to play a character. Your flavor shines through, and some people really enjoy that while comboing, instead of just in neutral or character selection.
Also, as a side effect of all that stuff, certain combo choices can be considered disrespectful, which further makes things better
A combo might be super long for just a hair more damage. Going for it, or ones in that style, consistently, is a sign that you either really really want to milk each neutral for all they're worth, or that you want to swag on them and make them wait and watch you do it. Or maybe you're consistently picking options that give them more meter when you should be making better choices, just so that they can burn it ineffectively and look the fool, or so that you're just throwing away free damage or meter for yourself. Or some combos allow for weirder enders that make actual taunts safe.
And as for what feels good, it honestly varies per person. Ganon is pretty trash in every Smash game, but his back air in PM just feels deeply good to me. That flick on the c-stick, that little fist popping back, that electricity when it hits. RARing into it also feels satisfying to do. It's also a decent move, but I love it more than I should because of the whole package—utility, execution, and audiovisual design.