...where to begin...
Baroque is French taken from the Italian 'barocco', meaning Bizarre. Believe it or not, a lot of the art that happened in the 'baroque' period (mid 16th century to mid 18th century) was considered bizarre. The term baroque, while generally associated with music, actually got it's start with a style of architecture beginning in German and Austria.
Of architecture... in a structural design sense, baroque refers to the emphasis of dramatic, and often highly strained effect and is typically exhibited by bold and elegantly curved forms, elaborate (sometimes to excess) ornamentation, and overall balance of disparate and dis-joined parts. This style was in usage from the mid 16th century to the late 17th century.
Examples of this philosophy can be seen mostly in the larger structures of the time, such as cathedrals (typified by ornate flying buttresses and grotesques) and royal residences (ornamental gateways and bulbous carved furniture).
Of music... The baroque music movement began at the height of the architecture movement, and lasted for about half-a-century after the building style ceased to be popular (late 16th to mid 18th).
Strictly speaking, baroque music follows the lines of being polyphonic and utilizing complex and elaborate music ornamentation and contrasting styles to the point of being a dramatic piece in and of itself.
Prior to this revival, the concepts of a 'solo', 'harmony' or 'concerto grosso' were not widely used. It was during this period that things such as symphonies, as we know them today, came about. Before this was the era of chamber music where each piece was small, elegant, to the point and most importantly, separate. In the baroque period we begin to see the idea of all pieces in a suite flowing into each other, almost as one gigantic composure.
Many of the old masters did the majority of their life's' work during this period: J. S. Bach, Antonio Vivaldi and Georg Friedrich Händel but to name a few.
It is sad to say, though, that after this period ended, many people began to use the term 'baroque' as a pejorative term to signify something as 'old', 'dated' or 'eccentric', when what is really should still mean is 'bizarre'.
Sources:
the Baroque Music Home Page (baroque-music.com)
Dictionary.com
Britannica.com