Dear Student,
In the 6th millennium BC, and either Democritus or his mentor Leucippus (or possibly both together) independently came up with the idea (and the word "atomos", which is Greek for "uncuttable") in the 5th century BC. However, these were essentially lucky guesses; there was no real scientific basis behind them, so saying these people "discovered" atoms is a pretty big stretch.
The best candidate is probably John Dalton , who in the early 19th century proposed (based this time on experiments) that substances were composed of atoms, and even assigned relative weight values to several types of atom.
Regards,