Hans Lipperhey of Holland is credited with inventing the telescope in 1608. However, given how long lenses had been around (by then, 2000 years), it is not likely his was the first telescope - he just generated more publicity. The problem was that the lenses of the time were so poor optically that the telescopes did little to improve the detail that could be discerned. Galileo took the concept and greatly improved the quality of the lenses to make telescopes that did allow much more detail to be seen, as well as gathering more light than the unaided eye could. Here is a modern test of the quality of Galileo's lens shown in the main notes:

The lens removed from its ornamental mount (left), from

http://brunelleschi.imss.fi.it/telescopiogalileo/etel.asp?c=50016

Test results on this lens. Evenly spaced and uniform straight lines would indicate perfect optical figure (right) - this lens would be pretty good in its center except for the discontinuity where it is broken, from

http://www.pacifier.com/~tpope/Interferometric_Data.htm

Despite his efforts to produce high quality lenses, the outer zones of Galileo's lenses were of such low quality that they degraded the images, and his telescopes all had stops that limited the apertures to about a centimeter (compare the centimeter scale in this picture). The combination of the lens flaws, the effects of the stop, and the characteristic that the lenses did not bring all the colors to the same focus all together limited the detail he could see to about 10 arcseconds, some 10 times better than the unaided eye.

from

http://brunelleschi.imss.fi.it/telescopiogalileo