Standard, Hardcore and Solo Self-Found
The two primary modes for playing Path of Exile are Standard and Hardcore. In Standard, if your character dies, you respawn at a nearby checkpoint and have the opportunity to try again. At higher levels you will start to lose a portion of your experience when you die. Don't worry, you can't go down in level from dying too much. In Hardcore, however, one death means you're out. Characters that die in Hardcore are sent to Standard, along with any items they were carrying. To reach a high level in Hardcore is a big achievement -- one that even very experienced players struggle with.
Both of these modes also have a Solo Self-Found (SSF) variant. In these SSF leagues players cannot team up or trade with any other players. That means the only items they have access to are the ones they've found or made. Players in SSF leagues can opt-out at any time and join the community of players in Standard (if they were playing Standard SSF), or Hardcore (if they were playing Hardcore SSF). There is no tangible benefit to playing Hardcore or SSF -- only the prestige. So if the idea of not being able to play with others or being booted out of your league if you die doesn't appeal, don't worry. It's the same game no matter how you choose to play.