What is the maximum level?

This answer is outdated as of the 2019 Commander Lilith DLC, as noted in this answer

The max level in the Borderlands 2 base game is 50, giving you 45 skill points to spend. Gearbox actually has a character planner on their website (which can be found here) that should help you pick out your skills.

Borderlands 2's "Ultimate Vault Hunter Upgrade Pack", available for $5 (or as part of the Season Pass DLC) can be purchased to increase the level cap from 50 to 61.

The Ultimate Vault Hunter Upgrade Pack 2: Digistruct Peak Challenge DLC further pushes the level cap to 72 for those who own it.

As of the release of the Commander Lilith & the Fight for Sanctuary DLC on 2019-06-09, the level cap has been increased to 80.