10 Thaats (basic scales) -> Ragas is the Hindustani music concept. In Carnatic Music there are 72 Melamkarthas Ragas yielding thousands of derivative ragas. (I limit my discussion to Carnatic ragas, hoping it'll answer your question)
At a basic level, each raga is a distinctive kind of melody, having its own flavour. A seasoned listener can detect this flavour and identify the raga, though he may not know the notes underlying; more or less like associating certain moods to Major and minor scales, or modes.
A bit more technical
The melody is not a single tune. One can identify a raga though the exact tune is unfamiliar. Technically, a raga has one or more scales (variants), and different kinds of embellishments/ornaments applicable for its notes under the circumstances, and certain characteristic phrases etc. All these are the devices which the musician employs to portray the mood of the raga.
Have a look at the scale of Dharbār Raga: sa ri ma pa dha ni sa - sa NI (long) dha pa ma ri GA GA ri sa
ascend: tonic, major 2nd, perfect 4th, perf. 5, maj 6, dim 7, 8th
descend: 8th, dim 7 (elongated), maj
6, per 5,
4, maj 2, min 3 (long), min 3 (again, long), maj 2, tonic
Notes in italics or oscillated in a particular way in this raga, ri (2nd) gets a bit of vibrato in the ascend, and there's Acciaccatura for some notes.
This video shows how to play the scale on violin:
Here you can find the sheetmusic (standard western notation) for a composition in this raga, and the video demo for the same: http://beautifulnote.com/blog/2010/03/dharbar-varnam-score/
another example: http://www.sarasvatibhavan.com/music_example.html
edit: This is Sri-Ragam, it has exactly the same tones (same 2nd 3rd
etc.), but a slightly different
ascend/descend. But it has a completely different mood & different set of ornaments.
more: In some ragas, a particular note is played slightly lower/higher than the usual pitch (microtones), in some other ragas such thing happen in select phrases. In some ragas some notes are played very rarely, some other notes are emphasised.
Raga vs. Scale
Western scales permit all kinds of permutations to get a musical phrase; one can combine notes in any ways they wish. In a raga, if a certain combination of notes, within its scale, doesn't follow the aesthetics of the raga, then it won't be permitted. However, a new raga can be spawned to accommodate such phrases. (There's a carnatic raga based of dheerasankarabaranam to imitate majorscale/chords like phrases)
Major, minor scales underwent centuries of harmonic exploration. Vertical relationships are studied and worked out. In raga music, consonance, dissonance relationships are worked out to construct melodies, but vertical relationships are not much explored.
- In the Major scale, 7th is called the leading tone, for its tendency to lead to the tonic. In Indian music, tendencies of notes undergo exhaustive study. For each raga, there are some stable notes (apart from tonic) on which on can pause, leading tones, special notes (like the 3rd & 6th in the above Dharbar example) etc.
Ragas can be best understood in the reverse order of technical aspects (ie., reverse of notes->scale->ornaments->phrases->raga) Some of these ragas existed for 1000s of years in folk melodies etc. The technique (scale for the raga etc) is merely an explanation. More like the theory of gravitation. Gravity existed well before the theory, and we have felt it. The theory just helps us to understand.