Short Answer
Tube amplifiers use vacuum tubes to produce their sound, solid-state amplifiers use transistor components to produce their sound. Each type of amp produces a distinct sound that has their own camps of fans.
Long Answer
Oh boy, one of the big questions, huh? Very well, then. Let's talk about amps. If you're entering the guitar amp market for the first time, you're bound to see some mention somewhere of "tube" amps and "solid-state" amps. There are some pretty substantial differences between tube and solid-state amplifiers. This all begins with tube amplifiers and their vacuum tube design and solid-state amplifiers and their transistor design. Vacuum tubes and transistors inherently create different sound profiles from the start, mainly based around vacuum tubes' ability to "break up" at louder volumes and (what most players would call) "warmth" when turned up.
Solid-state amplifiers' transistors are known to stay cleaner and tonally more consistent at varying volumes. This comes together as a greater strength for guitarists looking for a less volatile sonic palette that can operate at a greater range of volumes (think of the precision and cleanliness a jazz player might look for at louder volumes). Many solid-state amplifiers utilize a built-in distortion circuit to meet players half-way who are expecting breakup in their amp experience.
However you slice your amp experience, you've got plenty of options to choose from. The next chance you get, try out a tube amp and a solid-state amp back-to-back to come to your own conclusions and maybe even find a new favorite.