差点 usually implies the thing almost happened but didn't. For undesirable events, 差点 = almost (but didn't): 差点摔倒 (almost fell). For desirable events, add 没: 差点没赶上 (almost didn't make it, but did).
usage
差点 and 差一点 (chà yīdiǎn) mean the same thing. 差点 is the shorter, more colloquial form.
