Here's the most basic difference.
- First Person uses I: I went to the store. I bought a loaf of bread.
- Second Peron uses you: You went to the store. You bought a loaf of bread.
- Third Person uses he or she: She went to the store. She bought a loaf of bread.
What kind of story is better suited for each one?
That's entirely up to the writer. Whichever one helps tell the story the best.
In all three cases, it is entirely possible to get closer to the character by showing their internal monologue (thoughts). So it's not necessary to tell a story in first person to develop emotional attachment. Deep POV, or Limited Third Person, provides the reader direct access to that character's thoughts and emotions, as much as if told in first person.
For instance, in Jonathan Maberry's Patient Zero, all chapters told from the main character's point of view are written in first person. All chapters told from another character's point of view are written in limited third person (Deep POV). And it works very well. The reader has access to the thoughts and emotions of each character, but only to one character at a time per chapter.
You see this used in thrillers a lot because the main character can never know enough about the events to keep the story interesting. Often, the main character isn't even able to identify the villain until close to the end, so the writer has to give the reader multiple character points of view.
Are there advantages or disadvantages inherent to each form?
Not objective ones. It all depends on the writer and the story they're telling. I will sometimes write a section in two different points of view to see which one tells the story best.
For example, writing in the first person you are always following a character, while in the third person you can "jump" between story lines.
A writer should avoid jumping from one character to another within a logical unit. In most cases, this means sticking to a single character through a chapter, but could also work at the section or even paragraph level if the writer is skilled enough. This part of your question is more about character point of view (which character is telling the story) than point of view (first, second, third).
You rarely see pieces written in second person, but it's not unheard of: List of Notable Second Person Narratives at WikiPedia