Depending on the mower in question, things to check:
1) key in start position (if there is a key)
2) fuel selector not in bypass setting (if there is a selector)
3) safety handle properly depressed (if required)
4) primer bulb fully depressed specified # of times (if appropriate)
5) firm, quick, full pulls of starter cord
If it won't start after about 3 proper pulls, then it's time to get creative because it isn't you it's the mower.
Things to do at that point:
1) Is the fuel old? dump it start with fresh
2) Oil at proper level. Some mowers have a sensor that can tell the level.
3) As mentioned, did you flood it?
4) is the plug OK?
5) Check air filter. If suspect, attempt to start (But for God's sake don't mow with it!) with the air filter removed exposing the carb. Again, this is for test purposes only.
6) With air filter removed, apply a small amount of starter fluid. Just a spritz, yer not trying to spray paint with the stuff cause you'll ruin your engine with too much of it.
7) Remove bowel from carb. Check for debris and clean as needed.
Ah yes, Hubert got the one I forgot, and posted while I was typing - the choke setting. Very important, that
Those are the things I do when my small motors start giving me fits.