Dmitri Fyodorovich Karamazov, the eldest and half-brother to the other two, is accused of killing his father. He is sensitive and impulsive, very much in love for most of the book, a downright scoundrel at times but has the noblest sentiments of any character in the book.
Ivan Fyodorovich Karamazov, the middle child, is an atheist who struggles with his disbelief. He is the most intelligent and educated of the brothers, and he is the cause of two of the more interesting digressions, which are worth reading even if you don't read the rest of the book.
Alexei Fyodorovich Karamazov, the youngest in his early twenties, is a novice at the local monastery, and perhaps the most ambiguous of the brothers. He is faithful, but is not immune to crises of faith. He is the point of view character in most the scenes he appears in.
Alexei (Alyosha, Yoshka, or similar diminutives) is more often than not a conduit for the thoughts of other people, rather than a source of thought himself. Through most of the book he channels the wisdom of his Elder at the monastery, who dies fairly early on (relatively speaking), but this is also true symbolically, because he spends most of the book delivering messages to people. That being said, at the end of the book he recognizes the need for himself personally, and really shows his power of thought, of understanding, of empathy on the second last page. I really don't want to spoil the ending because it is worth reading the 800 tome just to get to the last page.
Alexei is also a point of stability in the lives of most of the important characters regardless of allegiance. He represents an ideal to most of the characters, of faith, hope and love, though he is far from perfect in these areas. It is the needs of others that pull him more into this role as things go on. On the elder's deathbed he tells Alyosha to go out and live among the people, rather than sequestering himself in the monastery, because of the real psychological need for him, almost as an extension of religion generally.