There are two main types of RPs, Open and Closed. The main difference between these are that in an Open RP you can jump in at anytime and in a Closed RP you have to sign-up or be invited in.
These types of RPs are open for you to come in and join at anytime. You will want to make sure your character is right for this RP and that you join in without disrupting the story already going on. Basically you don't want to bring your paladin into an RP full of undead working together to assassinate some because it won't make any sense to the plot. However bringing in your own undead could mesh right in. Also try to remember that if your character is just joining in try not do things like jumping into the middle of a fight that your character wasn't even near to begin with. Meeting up at their next location or catching up with them is always a great start.
Again, these are essentially the same as an Open RP but with one main difference: that you can't just jump right in. Closed RPs will have one person in charge that admits people into the RP. These types generally have a sign up thread for you to post in with what character you wish to bring in. Also if you see a Closed RP already in progress you can always send a private message to the person in charge of the RP and ask to jump in and they will usually give you a place to do so.
Generally you do not need a Bio written down and posted on these forums to join either type of RP. Sometimes people who start Closed RPs may require them though. It is always a good idea when first entering an RP that doesn't require a Bio that you make sure to describe your character fully so people can interact with you better. I mean you wouldn't want someone to come in and start interacting with you thinking you were a Blood Elf when you were actually a Goblin the whole time. Things like race and class have a huge impact on the way others interact with you.
Some Sign-up Closed RP Examples In the Sands... - A great example of all the different types of rules you can set down for your own RP. Tendrils of Nightmare - An example of how classes are limited on many sides of a conflict. The Cursed Tower - A excellent example of using background to set the mood of the RP and revealing the start of the story at the same time.
Starting a RP
Making your own RP is a bit more complicated. You have to create the story, choose the setting, fill in the background information and set any rules you feel nessecary. You also will be in charge of making sure the RP stays on track and people generally stick to the rules you set down (Mods can help you there). You also can choose to make it open or closed, who joins and if they need a written bio or not.
Generic RP Themes
Survival RP - You're stranded on an island, or lost in the jungle. How do you work together and survive? Horror RP - Essentially the plotline of any horror movie. A group of people are slowly picked off. Who will remain? Quest RP - You go on a quest, and meet people along the way who will help or hinder you. Inn RP - What's better than sitting at a bar, drinking ale, and listening to the chatter and music of an inn or tavern? Boarding House RP - A group of people live together. Usually with very comedic results. War RP - A war, with either the alliance vs horde, the alliance & horde vs a common foe, the alliance & horde vs a common foe, while the horde and alliance subtly try to undermine each other, so one comes out of the battle on top, or even custom factions pitted against each other, a common foe, the alliance, or the horde. This is generally a SRP. Powerful characters (ex: Dragons) must be neutral in the conflicts, either as a bystander or as someone who does not do ill to the horde or alliance (its okay to help destroy the common foe though (the horde and alliance will most likely need the help)