If you have come across this issue, do the following:
1) Check the Minecraft skins server status. If it's down, it's the cause of the issue. In this case, only Mojang can fix this issue.
2) Disable the Nick Hider mod. This mod has an option to hide players' skins.
3) Try re-logging on the Vanilla Minecraft launcher.
4) Make sure you don't have installed software that is incompatible with Minecraft that causes login and multiplayer connectivity problems.