Telescopes can generally be divided into three categories based on their optical design:

• Refractors (dioptrics) - These use only lenses to refract (bend) the light into focus
• Reflectors (catoptrics) - These use only mirrors to reflect the light into focus
• Catadioptrics - Use a combination of lenses and mirrors to both refract and reflect the light into focus

Within each category are many variations, each with it's own advantages and disadvantages as they were improved on over time. But, the majority of the telescopes you'll encounter these days are likely to be one of only a handful:

• Refractors
  • Achromatic
  • Apochromatic
• Reflectors
  • Newtonian
  • "Dobsonian"
• Catadioptrics
  • Schmidt-Cassegrain
  • Maksutov-Cassegrain

For a complete list, see Wikipedia's "List of telescope types" article.