Normalizing is taking the track and adjusting the level so a chosen part of the track hits a defined level. Peak normalization uses the highest peak in the track as the reference point. RMS normalization uses the average RMS level of the track as the reference. So if your mix has a few dB of headroom left and you want to make it as loud as possible, you can peak normalize to 0 dBFS and it will boost the level up so that the highest peak on your track will hit 0 dBFS. Doing that is a fast way to boost the signal as much as possible with no clipping and no changes to dynamics.
Limiting is different. That is actually squashing any signal that would go above, down to a set threshold. Limiting is basically compression with an infinite ratio. In a sense, clipping is a very crude form of hard limiting.