Extrasolar planets are planets that orbit a star that is not our sun. They are often lightyears away and astronomers study them using spectroscopy and other methods to determine where the planets are.
Space weather is how the environment around the Earth and solar system may vary because of activity at the sun. The Northern Lights is a harmless example of space weather. It is an effect of an event called a Coronal Mass Ejection or CME.
The crew of the Enterprise have traveled back in time from the 24th century to make sure that a pivotal moment in their history occurs, the warp drive rocket launch that would lead to human's first contact with aliens. They find the inventor of warp drive and captain of that crucial launch, Zephraim Cochrane, and make him realize the importance of his role.