STAB (Same Type Attack Bonus) is a 50% boost to a move's power when the Type of the move is the same as the Type of the Pokemon using the move. Whatever the power was before, it gets multiplied by 1.5 or 150%.

e.g. If the base power of the move was 90 and it's used by a Pokemon of the same Type, it becomes 90 x 1.5 = 135.

And to clear up any confusion, yes, Normal-type Pokemon also receive STAB when using Normal-type attacks.

Dual-Type Pokemon

Pokemon with Dual-Types get STAB for both Types. For example, Dialga, a Steel/Dragon Pokemon, will receive STAB for both its Steel-type attacks and its Dragon-type attacks. This can make for some interesting, and powerful, moveset options.

STAB and Other Power Boosts

All power boosts are cumulative. So, if you have a Fire-type Pokemon using a Fire-type attack while Sunny Day is in play, you get the x 1.5 boost for STAB, which then gets multiplied by another 1.5 for the Sunny Day bonus. 1.5 x 1.5 = 2.25, so your move power more than doubles.

e.g. If Charizard uses Flare Blitz (base power 120) while Sunny Day is in force the new power of Flare Blitz becomes 270.

Things get really interesting if your Charizard's HP are below 1/3 of the maximum, because then its Blaze ability kicks in. This provides Charizard yet another 50% boost. So with Sunny Day + Blaze + STAB, the new power of Flare Blitz becomes 405! Of course, Flare Blitz has recoil damage, so your Charizard with less than 1/3 maximum HP would proceed to launch one ridiculously powerful attack and knock itself out. But it would go down in a Blaze of glory! (sorry ;D)

Hidden Power and STAB

If a Pokemon's Hidden Power is the same as their Type, they will receive STAB for it. Example, if a Flying-type Pokemon has a Flying-type Hidden Power, it will receive STAB when it uses the attack. Always remember to go by the actual Type of the Hidden Power when thinking about STAB, not the fact that the games list it as being a Normal-type attack. Since a Hidden Power can never be a Normal-type, you can't ever get Normal-type STAB from it.