All of these mods are designed to work with Forge, the most popular modloader for Minecraft.
Now, here is a list of mods that work in multiplayer Minecraft.
These three mods, all by the same author, are great for Minecraft PvP. Armor Status adds armor icons in the corner with durability healthbars. This can alert you if you armor is going to break in the middle of a PvP fight. Direction HUD adds a compass-like display on the screen that can tell your direction, and Potion Effect HUD lets you view your potion effects without having to open up your inventory. This is useful if you are in a battle and need to know when to drink another regeneration potion, for example.
This handy mod adds a map to the corner of the screen that can show you the terrain, nearby mobs, or even other players in your area. It's very customizable, and you can toggle it on and off. It's great for Minecraft PvP, because it makes it harder for players to sneak up on you and catch you unawares.
This mod surprised me! It allows you to see the health of nearby players and mobs without any kind of server plugin. This is extremely useful in Minecraft PvP because it allows you to see how much health your opponent has mid-battle. To see a the health of a mob or player, just look in it's direction.Also, if you don't like the default skin of the damage health bar, you can change it by downloading other skins on the mod's page.
Optifine really is a must if you have an older computer or if you just want to maximize your FPS. This mod allows you to use HD texture packs, allows you to have much more control over visual settings, and adds a zoom-in feature by pressing the left control key.
Keep in mind that while all these mods are usually allowed on servers, some servers don't allow players to use them. If you're worried that you may be banned for using them, first keep in mind that these mods are impossible for server admins to detect. But to be on the safe side, read server rules or ask an admin. I'm not responsible if you get banned or otherwise punished for using these mods on servers where they aren't allowed.