Make sure that your gamemode is not set to adventure mode.
- for regular survival mode enter :
/gamemode <playername> 0
- for creative mode enter :
/gamemode <playername> 1
- for adventure mode enter:
/gamemode <playername> 2
- for spectator mode enter:
/gamemode <playername> 3
Each mode acts a bit differently
- Survival mode is just the Minecraft as everybody knows it and this is probably the mode you want to be changing in.
- Creative is the mode in which you have unlimited blocks and resources to build everything in your imagination. You also have the ability to fly.
- Adventure mode is the same as survival only you dont have the ability to destroy blocks (without the proper tool).
- Spectator mode allows you to go inside players and mobs, as well as go through world borders and go down below the surface without getting suffocated (unless you change your gamemode).
If you know you're in the correct gamemode, another possibility is that you are too close to the spawn. This typically only applies to servers if they have some sort of spawn protection enabled.
Also clicking does not break (many) blocks, you need to click and hold. This doesn't apply to Creative mode. Initially only a few block types can be feasibly broken using only your hand, including dirt/grass, sand, wood and leaves, so try left-clicking and holding and see if you can break these types of blocks.
If nothing is happening when you left click, then your mouse button might not be properly mapped. Go into
Options... -> Controls... and change Attack to your mouse button, typically Button 1. (Just click twice to change it.)
Finally, if you're playing on a server, there may be synchronization issues preventing you from breaking blocks, or performing other actions. This is typically referred to as "block lag", and is caused by the server not being able to keep up with updates coming from players. There isn't much you can do about it, unfortunately.