What would happen if 90% of the Bitcoin miners suddenly stopped mining? What would happen if 90% of the Bitcoin miners were to suddenly stop mining Bitcoin, i.e. the Bitcoin network's hashing power would drop to 10% of the previous amount. 

Would it become more difficult for the rest of miners to mine? Would transactions take longer than usual?
The probability of 90% of miners suddenly stopping mining bitcoin is only slightly more probable than the chance of the sun going dark this afternoon.

Anyway, supposing that 90% of the active bitcoin miners were to suddenly vanish, the first effect would be that blocks would take much more time to be confirmed (10x longer, so 100 minutes per block instead of 10 on average). The next time the difficulty is adjusted (up to 2016 blocks later), the difficulty would be reduced such that blocks again take about 10 minutes to confirm depending on the new speed of the network as a whole. The difficulty is normally adjusted so that 2016 blocks is about two weeks, but if you just lost 90% of your hashing power then the next adjustment could be up to twenty weeks later.

The last time the difficulty was 10x less than it is now was around October 2013. You can see the difficulty history here .