So far you have 5 people claiming the throne: [These are not spoilers, just reminding people who havent read the books where we are as we go into season 2]
Stannis Baratheon: The oldest of Robert's brothers, so given what we know about Joffrey being a bastard of Cersei and Jaime's incest (but only a few people on the show know it though several suspect it), Stannis is the rightful heir to Robert.
Renly Baratheon: Youngest brother of Robert and Stannis, not the rightful heir, but thinks Stannis is more of a fighter than a king, so Renly makes a claim to the throne. Renly is more popular and charismatic than Stannis and has the support of the powerful Tyrell family (second only to the Lannisters in Westeros) because of his ties to Loras Tyrell (the knight Renly has a hidden gay relationship with).
Joffrey Baratheon: Thought by most of Westeros to be the true heir to Robert since he is Robert's oldest son, but of course we know he is not Robert's son at all, and is really a Lannister bastard, so his claim is not true. Plus he's an evil, sadistic, spoiled punk. On the other hand, right now he actually holds the Iron Throne.
Robb Stark: Ned's son, recently named the King in the North. He is only claiming the north, not the iron throne over all of Westeros, but in the end he will have to defeat the Lannisters and take over all of Westeros if he wants to remain king because the Lannisters are not going to just let him have the north.
Dany: Dany probably has the best claim to the throne, but of course she is not even in Westeros and has lost her Dothraki husband Drogo and the Dothraki army she hoped to use to take over Westeros. On the other hand, she has just "given birth" to three dragons as season 1 ended, which are the first dragons to exist in the last 100+ years.
Here is some more background info that will be relevant heading into season 2. There are no key plot spoilers, but I am talking about some new characters that are being introduced this season, so I will spoiler tag it for those who don't want to hear any info like that:
Spoiler!
There will be a 6th person claiming the throne, it will be Balon Greyjoy from the Iron Islands just off the coast of Westeros. He is Theon Greyjoy's father, the one who rebelled against Robert years before, but was defeated by Robert and Ned Stark, which is how Ned wound up with Theon as his ward to prevent future uprisings. He will be introduced in season 2.
Some other characters being introduced this season are Brienne of Tarth who is a not very attractive female soldier who loves Renly and serves him. Stannis Baratheon's right hand man, Davos Seaworth, is also introduced. He is honest and loyal to a fault. Finally, there is the Red Priestess, Melisandre, who "guides" Stannis and Stannis' wife.
There is some conflict in religions in this show as the northmen tend to worship the "old gods" and pray to the weirwood trees.
In the south of Westeros they tend to worship "the seven" which are the mother, the father, the stranger, the warrior, etc., Ned's wife worships the seven.
Melisandre and the red priests worship Rholor (not sure on spelling) the god of light, and the red priests can see future visions in their flames (though they are often hard to interpret), and sometimes have other special powers and abilities.