A short name would be an IDC connector, here's a whole list of them:
http://www.jst.com/home21.html
This one looks close to what you want:
http://www.jst-mfg.com/product/detail_e.php?series=246
Just measure the pitch (pin to pin spacing) and you should be able to find a mate. If I remember right, finding those in the US at that pitch and in 1-100 unit volumes was a pain. JST was eventually the guys who had what I was looking for though.
DIL / dual-in-line header.
These are available with insulation displacement connectors (IDC) allowing termination to ribbon cables, and can be bought preterminated. Molex make these and also probably have solderable versions (you'll be sorry).
IF the pitch is 2.54mm, which seems likely, then an old floppy disk drive cable (34 pins) is probably an easy solution.
You MAY have a 2mm pitch header but 2.54 mm or 2.5mm is much more likely.
Spacing is usually either 2.54mm = 0.1" or 2.5mm = not quite 0.1".
The two are interchangeable (for non-purists) after a fashion for small numbers of pins but get out of alignment by 0.04mm per pin.
When measuring something like this it is a very very (very) good idea to give people as accurate a pitch measurement as you reasonably can. In this case you have 16 pins so there are 15 x pitch distances between 1st and last pin. Measure this distance and divide by 15. Or measure 10 pitches between pins 1 & 11 and divide by 10.
Even for 10 pins 10 x 2.5 = 25mm and 10 x 2.54 = 25.4mm so you could probably detect the difference with a standard metal metric ruler calibrated in mm. At 15 pitches it is 37.5 and 38.1 mm for a difference of 0.6mm. OR if it is some pitch closest to 2mm (unlikely) it will be different.
2 of these](http://www.pololu.com/catalog/product/973) from Polulu may do, depending on how they butt, BUT you can buy 32 pin ones.
An old floppy disk drive cable is 2 x 17 pin.
A HDD parallel access cable is 40 pin
Various examples here
Mouser have a very nice parametric selector that includes cable assemblies BUT no 32 way flat cables listed. (Check for yourself just in case).
Best Answer
That's a DIN 41529 loudspeaker connector.
It's simple, it's polarized, so it was only a matter of time the Chinese used it for LED lighting power connectors.