Excluding translucent pieces, how many colours of official LEGO bricks are there?
The official LEGO color chart at Peeron has 111 colors listed. There are 14 listed as "transparent" (or "tr"), so - 97.
The Peeron guide also lists CMYK, RGB, HTML Hex Codes (displayed as the sample), and Pantone numbers for each.
The latest palette officially released by LEGO in 2010 has 51 colors in use, including 33 regular colors, 14 transparent, and 4 "effect" (silver, gold, metallic, phosphorescent).
Of course, as 81bronco points out, there have been a lot more, but these are the one LEGO was effectively using as of January 2010. In the few years before that, LEGO was busy rationalizing its palette of colors to eliminate seldom used colors as much as posible. Considering the downsize to 51, I don't think they've removed any more since.
This Flickr image from Brick Colorstream shows all the known colors of LEGO bricks, charted along a timeline of when they were available. According to that image, there was one new color in 2015, "Conductive Black".
According to the LEGO customer service site which allows you to search by color family, there are 9 color families and 23 exact colors:
Color Family Exact Color ------------------------------------------- Black BLACK Blue BRIGHT BLUE Blue EARTH BLUE Blue MEDIUM BLUE Blue TR. BLUE Blue TR. LIGHT BLUE Green BRIGHT GREEN Green BRIGHT YELLOWISH GREEN Green DARK GREEN Green TR. FLUORE. GREEN Green TR. GREEN Grey DARK STONE GREY Grey MEDIUM STONE GREY Orange-brown BRIGHT ORANGE Orange-brown DARK ORANGE Orange-brown FLAME YELLOWISH ORANGE Orange-brown REDDISH BROWN Orange-brown TR. BROWN Purple BRIGHT PURPLE Purple MEDIUM LILAC Red BRIGHT RED Red NEW DARK RED Red TR. FLUORE.REDD. ORANGE Red TR. RED White WHITE Yellow BRICK YELLOW Yellow BRIGHT YELLOW Yellow SAND YELLOW Yellow TR. YELLOW