Why were Luke and Leia split up and given to two different families?

In-movie logic: Darth Vader will be looking for twins, not single children, and splitting the pair makes it more likely that one would remain free in case Darth found the other. What is less logical is that Obi-Wan selected as foster parents couples known to Anakin/Darth rather than some complete strangers. Maybe he decided that Darth would think it unlikely that he would do something that stupid.

Out-of-movie logic: When the first movie was made, George Lucas had not yet decided that Luke or Leia were siblings, nor that Darth Vader was their father. The latter was decided during the making of The Empire Strikes Back (it is not in the original script by Leigh Brackett, where a force ghost of Luke's father shows up). That the two are siblings wasn't decided until the making of Return of the Jedi - when Obi-Wan says to Yoda, "Remember, there is another" in TESB, Lucas had not yet decided who that "other" was.

This is addressed in the film's official novelisation. Obi-Wan suggests that they be split up as allowing the discovery of both (at the same time) would be an intolerable error on their part.

“We should split them up,” Obi-Wan said. “Even if the Sith find one, the other may survive. I can take the boy, Master Yoda, and you take the girl. We can hide them away, keep them safe-train them as Anakin should have been trained-“

In the film the dialogue is reversed (Yoda suggests they be split up without explanation) but I think we can be reasonably confident that the reasoning is much the same.

Because they wanted to avoid the Sith sensing their presence

Remember, Anakin was born off the Force; his powers were not what the Jedi expected to be. Qui-Gon even says that Anakin's midi-chlorian count is more than that of Master Yoda.

OBI-WAN : Even Master Yoda doesn't have a midi-chlorian count that high!

... [later]

QUI-GON : A boy... his cells have the highest concentration of midi-chlorians I have seen in a life form. It is possible he was conceived by the midi-chlorians.
_{Star Wars Episode 1: The Phantom Menace}

Yoda, after the birth of the twins suggests to split them up because of the very same reason. He thinks that the twins might attract much more attention together due to the affinity to the Force. And at the same time, if one is discovered, the other is still safe. As Obi-Wan Kenobi says,

YODA: Hidden, safe, the children must be kept.

OBI-WAN: We must take them somewhere the Sith will not sense their presence.

YODA: Hmm. Split up, they should be.
_{Star Wars Episode 3: Revenge of the Sith}

As to why Leia (and not Luke) was taken to Alderaan, Senator Organa requests so.

BAIL ORGANA: My wife and I will take the girl. We've always talked of adopting a baby girl. She will be loved with us.

OBI-WAN: And what of the boy?

YODA: To Tatooine. To his family, send him.

OBI-WAN: I will take the child and watch over him.
_{Star Wars Episode 3: Revenge of the Sith}