Karate: Lots of focus on Katas (pre-arranged "forms" of movements. Some sparring, but it's mostly low contact "point" style sparring, with protective gear (Kyokushinkai is one of those exceptions to the generalization here, as they spar full contact - until knock down or knockout, with no protection, but with no punches to the head). Lots of time doing traditional stances, kicks, reverse punches, etc. I'm not so sure this is very effective in self defense situations, as many of the techniques are very ritualized, and unrealistic.
Tae Kwon Do: Same as Karate, except much greater focus on high kicks. Even less realistic for defense than Karate.
Japanese Ju-jitsu: Focus on stand up grappling, wrist locks, escapes from grabs and chokes, many throws and falls. Has the same ritualized formal practice as Karate - lots of practice against an un-resisting, unmoving attacker than grabs and then freezes, or punches then freezes. Some punches and kicks thrown in, but just a few. I'm not sure this is so great against a resisting, moving opponent - mainly because the practice methods don't allow practice in that sort of situation. The techniques are more realistic than Karate, but I'm not so sure all of them are usable in real life. Since you don't have to punch or kick to fight or repel and attack, this is ideal for use against someone you don't want to hurt - a drunken uncle at a party for instance. Against a determined attacker, I feel some of the primary techniques are less useful.
Aikido: Similar to Japanese Ju-jitsu, but more focus on flowing with an attackers force. More philosophy, less "hard" techniques. I think most of the criticisms of Japanese Ju-jitsu apply here as well.
Hapkido: About 80% Tae Kwon Do, and 20% Ju-Jitsu. Same observations of those arts apply.
Judo: Has many of the techniques of Japanese Ju-Jitsu, but has many of the "deadly" techniques removed. By taking out the punches, kicks, wrist locks, etc. you are left with some techniques that are safe to be used in a sport. Lots of throws, falls, trips, and groundwork. Most of the practice is against a resisting, moving opponent.
If you can throw someone who is expecting it, and has trained in resisting it who is REALLY trying to stop you - then the odds are really good you can throw some guy in a fight who doesn't expect it. Its weakness is that it is missing striking, and maybe some of the "deadly" techniques that might really be useful.
Western Boxing: Since you are practicing against someone actively resisting, as in Judo - the training has great value for actual self defense. The weakness is that only the hands are used, there is no grappling or kicking. Also, much of the body isn't a valid target. It's also worth saying that the transition from wearing gloves (and fighting someone wearing gloves) is a big one.
External Kung Fu: There are many styles that fall into this category, and there are some big differences between them. There are ones that focus on punching and kicking, and others that include techniques similar to Ju-Jitsu (or you could say Ju-Jitsu has techniques similar to them). It's worth pointing out that Karate is basically a Japanese version of an Okinawan version of certain external Kung Fu styles.
There are forms as in Karate, and some schools practice sparring as in Karate. The techniques are a bit different of course. Kung Fu in general is a bit "softer" with more circular motion, and flowing stances, versus the linear movement of Karate. Of course, that's a generalization - some styles are very similar to Karate. Some styles such as Wing Chun focus on "trapping" which involves hampering and controlling an opponents arm so as to facilitate striking, and to prevent them from striking you. Other arts, such as Chin-na focus on grappling, pressure points, and locks as seen in Japanese Ju-jitsu. I feel that many of the same critisms of Karate and Japanese Ju-jitsu apply here.
Internal Kung Fu: Tai Chi Chuan is the best know style in this category. The other styles are Hsing-I, and Ba Gua. The focus is on training the generation of Qi (life force or energy), and the application of Jing (the manifestation of power through the use of Qi). The idea being that this energy is generated internally to the body, rather than through gross muscle movements as in Karate. Practice involves slow practice of forms, drills such as "sticky hands", and basically pushing a partner. There are punches, kicks and other strikes of course - but the emphasis is on the power coming from the Internal methods. There isn't much sparring that I have seen. It would seem that these arts don't have a lot of application in actual self defense.
Muay Thai: This art has western style boxing punches, elbow and knee strikes, and kicking. This is practiced as a sport, so many of the positive comments related to boxing would apply here as well. In my opinion this style has the most effective kicks of any
Style - utilizing a very powerful round type kicks to the leg as a common target area.
There is also a significant amount of techniques done from trapping or clinching range.
The disadvantages would be that there are very places to train in this art in the West, and the training is very strenuous and hard on the body. You probably won't find a school that trains for actual Thai style matches, but rather a school in another art that incorporates the techniques specific to this style. There is a lack of grapping beyond the clinching techniques I have mentioned.
Brazilian Ju-Jitsu: This is derived from Japanese Ju-jitsu, but is actually more similar to western wrestling, with elements of Judo and Ju-jitsu. Western style wresting takedowns are used, and most of the focus is on grappling on the ground. Chokes, armbars, and submission holds are used. The training is against a resisting opponent, as in boxing.
One disadvantage is that the majority of the training is on getting an opponent on the ground and fighting them there. It's very debatable whether this is advisable in most environments that self-defense scenarios would occur.
I purposefully have left out topics such as using weapons, defending against weapons, and fighting multiple opponents. These are all very much worth discussing, but are beyond the scope of what I'm discussing here. However, these topics are worthy of serious consideration.
I also haven't mentioned non-self defense benefits of these arts. That was intentional as well, but it's worth noting, that the non-self defense benefits probably far outweigh the self defense benefits for most people. If you aren't a bouncer, repo man, cop, or someone who is in dangerous situations on a regular basis - concerns such as health benefits, fun, and cost probably will be more important than self defense.