The Islamic State is a jihadist group that is currently seizing territory in Syria and Iraq. They are one part of the cause of the Syrian refugee crisis. Like Deborah said, their goal is to establish a state with Sharia law.
The Taliban is currently an insurgent group in Afghanistan. They were previously a militia and then in control of the government in that country. Like IS, they follow a very strict, anti-modern interpretation of Sharia law and have been engaged in violent conflicts in Afghanistan since the 1990s.
Al-Qaeda is a global terrorist network/organization. They have been involved in multiple conflicts since 1988, and have played a role in Afghanistan, Iraq, and Syria which is why all three of these often get mixed together or brought up when discussing certain topics.