I watched Star Trek The Next Generation a few minutes ago and I noticed a character named Wesley Crusher but everyone calls him Wes for short and I would like to know if Wesley Crusher is Dr. Crusher's son? And was Wesley Crusher a character only seen in the first season of Star Trek The Next Generation or did he appear as a regular character throughout the series's run?

Yes Wesley is Dr. Crusher's son. Wil Wheaton was a regular the first four seasons, then was gone when Wesley went to Starfleet Academy.

So Wesley Crusher is indeed Dr. Crusher's son on Star Trek The Next Generation. I thought Wesley was smart but sometimes his inventions would get him in trouble but Dr. Crusher seemed to be a great mother to Wesley