Note: This article handles Assault Marines - as in the Fast Attack choice in Codex: Space Marines. This article may in some ways not apply to the Assault Marines in Codex: Dark Angels, Codex: Black Templars and the Blood Angels update or the Blood Claw "Bikers" entry in Codex: Space Wolves. The set-up may be different due to other prices of wargear, but the tactics are more or less the same.
The Assault Marines is one of the true CC-dedicated units in the Space Marine codex. Since the regular Assault Marine will get 3 attacks on the charge with MEQ WS and S, and they all have Power Armour to protect them, they are able to do quite some damage, and take a beating in return.
They are - however - sadly outmatched by any CC-specialist any other army has. Wyches, Harlequins, Genestealers and so on will kill them in close combat if you're not careful. They are faster than you, and they will in a lot of cases ignore your armour save. Thus, I will give you some advice for using these flying Space Marines to the max.
Generally, assaulting squads wants to be big, and this is true for the Assault Marines as well. 8 or 10 men is ideal when it comes to survivability, victory points and punch in close combat. So if you've got room for it, take 10 of them, if you don't take 8 - or none. The squad has access to Flamers, which is nice considering their speed - they will be able to position themselves neatly for flaming Gaunts, Boyz, Kroot, Guardsmen and other light infantry with devastating effect. But against other MEQ armies, which is the most likely opponent for your Assault Marines, the Flamer will have little effect. Also, the bearer will lose the extra attack for being armed with two one-handed weapons.
So, for the most versatile squad available, take two Plasma Pistols. Plasma Pistols can deal with almost anything - ranging from light infantry to heavy infantry to lighter tanks, such as transports and skimmers. Due to their speed, Assault Marines can shoot tanks in the side or even the rear if they are positioned ideally, and this makes the Plasma Pistols deadly. They are also cheaper than Flamers.
The Sergeant should always be upgraded to a Veteran Sergeant and carry a Power Fist. Hidden power fists are great in all aspects, they are hard to avoid, cheap and can take down just about anything. No Plasma Pistol on him though, since he's already expensive - and he can burn himself if he's unlucky.
There are also two options that aren't exactly mandatory, but can be viable if you've got the spare points. Meltabombs can be taken quite cheaply, and will provide the squad with a good punch against vehicles, other than the Power Fist on the Veteran. Take them if you face a lot of tanks and they won't disappoint you. If you've selected either of the traits No Mercy, No Respite; or Blessed be the Warriors, the Assault Marines can have the veteran skill Furious Charge. And now we're getting to the good stuff. Furious Charge makes the unit very, very potent in assault. That one I recommend strongly if you haven't selected your traits yet.
For a truly devastating set-up, add a Chaplain with a Jump Pack. The Litanies of Hate special rule will make your Assault Marines deadly. Just give the Chaplain a Bolt Pistol, a Jump Pack and Frag Grenades and you are good to go. The extra Power Weapon carried by him also helps out against a lot of enemies.
The only composition I do not recommend is to remove the Jump Packs for one or another reason. Assault Marines should generally always have Jump Packs.
Assault Marines can be used for two things generally: either they form some kind of bulk of your army and assault as soon as possible - in which case you'll need at least two of them, or they are used for counter-charge in a shooting army, and then you'll do fine with one. If you take two or more squads, I'd recommend you to have them at 8 men a piece, since they will be very expensive otherwise.
When using the counter-charge squad, simply keep it out of line of sight until enemy troops are close enough to charge. Remember, softening up a target of CC-specialists of your enemy is always a good idea - especially if they carry something that ignores armour saves, rends or lower the Assault Marine's save.
If you plan to take two or more Assault Marine squads and rush them at your opponent, to get them locked in combat as soon as possible, Furious Charge is almost always needed. They also need cover in some form to be able to reach the enemy lines without getting shot up. To the contrary of a lot of non-Marine players belief, Power Armour is not the equivalent to a tank when it comes to protecting your troops. They will die horribly if you keep them in the open, either to massed fire from low to mid strength weapons, or from weapons with fewer shots but good S and AP. Cover is always your friend, use it to the maximum.
Avoid area terrain at all costs, as you are almost bound to roll one or two ones for your 8 or 10 Marines. Keep them behind woods, but not in them! Also remember to use any form cover, not just the kind that blocks Line of Sight. Cover saves can be a life saver against a lot of weapons, so barricades and brick walls are your friends.
The best way to deliver your Assault Marines into your opponent's lines is to use Rhinos, and rush them ahead to shield your troopers behind. The first turn, you move up the Rhino 12" and pop smoke, and you follow right behind with your Assault Marines. The next turn, you move 12" again, disembark the passengers (Tactical Squads in most cases) and you have them shoot at the biggest CC-threat in front of them. The Assault Marines then charge in at the most dangerous shooting unit in front of them, and rends them into shreds.
To summarize, make sure you have the correct set-up for your Assault Marines so that they have good synergy with the rest of your list, pick your fights carefully and utilize cover to the maximum, and your Marines will do well on the battle field.