Minecraft wiki entry for Defeating_a_Nether_fortress, associated text: Find the natural regeneration unable to keep up with the fights you get into? Consider Regeneration potions, made with Ghast Tears dropped by Ghasts in the Nether, and enhance then with glowstone dust (fast healing) or redstone dust (heals you for a while)
Need extremely quick heals between fights and can't wait for Regen to help? Use Instant Health potions, made with Glistering Melon Slices (made with gold and melon slices), for this. Glowstone will enhance the effect to more healing.
Is even sprinting not fast enough for you? Speed is useful. It uses sugar, a fairly common find, to give you the speed effect. Enhance it with either glowstone dust (speedy) or redstone dust (speedy for longer)
Your sword just not doing enough damage? Strength can help. It increases your melee damage by 3 health points normally, 6 if enhanced with glowstone dust. Redstone dust can also be used, giving you the extra strength for longer.
On the flip side, "Corrupted" potions also exist. They inflict harmful effects (like Slowness, Weakness, and Instant Damage to name a few). Take almost any beneficial potion and brew a fermented spider eye in, and the effect will become corrupted. You normally wouldn't want to drink these sorts of things, which is why splash potions exist.
Splash potions allow you to inflict harmful effects on your enemies (see above) (or beneficial effects for allies). Brew gunpowder into any existing potion and now you can throw it. Make sure to throw it far away enough not to effect you when it lands for harmful potions. Make sure to throw it at groups of enemies, as launching it at one would be wasteful.