Ragas are the melodic modes used in Indian classical music. "Raag" is the modern Hindi pronunciation used by Hindustani musicians; "Raagam" is the South Indian form used by Carnatic music musicians.
Description
A raga functions both as description and prescription. It describes a generalized form of melodic practice; it prescribes a set of rules for how to build a melody. It specifies rules for movements up (aarohanam) and down (avarohanam) the scale, which notes should figure more and which notes should be used more sparingly, which notes may be sung with gamaka, phrases to be used, phrases to be avoided, and so on.The result is a framework that can be used to compose or improvise melodies, allowing for endless variation within the set of notes.
Although notes are an important part of raga practice, it by no means exhausts what a raga is. A raga is more than a scale. Many ragas share the same scale.
The underlying scale may have five, six or seven tones made up of swaras. This provides one method of classifying ragas. Ragas that have five swaras are called audava ragas; those with six, shaadava; and with seven, sampoorna (Sanskrit for 'complete'). Those ragas that do not follow the strict ascending or descending order of swaras are called vakra ('crooked') ragas. (To see the order of notes, check the article on swara.)
The basic mode of reference is that which is equivalent to the Western Ionian mode (this is called Bilawal thaat in Hindustani music and shankarabharanam in Carnaitc music). All relationships between pitches follow from this basic arrangement of intervals. In any given seven-tone mode, the second, third, sixth, and seventh notes can be natural (shuddha, lit. 'pure') or flat (komal, 'soft') but never sharped, and the fourth note can be natural or sharp (tivra) but never flatted, making up the twelve notes in the Western equal tempered chromatic scale (but without Western pitch equivalencies like, for example, A# and Bb). A Western-style C scale could therefore theoretically have the notes C, Db, D, Eb, E, F, F#, G, Ab, A, Bb, B. Ragas can also specify microtonal changes to this scale: a flatter second, a sharper seventh, and so forth. Treatises from the first millennium report that the octave used to be divided theoretically into 22 microtones ("shrutis"), but by the 16th century, this practice seems to have died out. Furthermore, individual performers treat pitches quite differently, and the precise intonation of a given note depends on melodic context. There is no absolute pitch; instead, each performance simply picks a ground note, which also serves as the drone, and the other scale degrees follow relative to the ground note.
Some Hindustani (North Indian) ragas are prescribed a time of day or a season. During the rains, for example, many of the Malhar group of ragas--associated with the monsoon--are performed. Some musicians take these prescriptions very seriously. However, since the majority of concert hall performances take place in the evening and night, musicians often have to make concessions for the sake of public performance.
The two streams of Indian classical music, Carnatic music and Hindustani music, have independent sets of ragas. There is some overlap, but more "false friendship" (where raga names overlap, but raga form does not). In north India, the ragas have recently been categorised into ten thaats or parent scales (by Vishnu Narayan Bhatkhande, 1860-1936); South India uses a somewhat older, more systematic classification scheme called the melakarta classification, with 72 parent (melakarta) ragas. Overall there is a greater identification of raga with scale in the south than in the north, where such an identification is impossible.
Note that the term "parent scale" is a metaphor, and is potentially misleading. It might seem to imply that scales came before ragas, or that ragas are made from scales. In fact, it's the other way round--parent scales (both melas and thats) were induced from raga practice. Again we stress that ragas are not scales.
As ragas were transmitted orally from teacher to student, some ragas can vary greatly across regions, traditions and styles. There have been efforts to codify and standardize raga performance in theory from their first mention in Matanga's Brhaddesi (~10th c.) Some people approach raga performance from the Vedic philosophy of sound; others from a Sufi perspective; still others approach raga primarily as an aesthetic entity; others approach it as a kind of combinatorics.
Indian classical music is always set in raga, but all raga music is not necessarily classical. Songs range from being clearly in one raga or another to being in a sort of generalized scale. Many popular Indian film songs resemble ragas closely. Again, it is important to stress that just even if song shares a scale with a raga, it isn't necessarily "in" the raga.