Skyrim's console is a developer tool that shipped with the game. It is not a cheat menu, and it does not stop you doing something destructive. That power is the reason it is useful and the reason it deserves care.
Opening the console
Press the tilde key — ~ — during gameplay. The game pauses while the console is open. Type a command and press Enter. PC only; there is no console on any console version.
The essentials
tgm — god mode. Blocks all damage and removes the limits on stamina and magicka. Toggles off with the same command.
tcl — no clipping. Walk through walls and terrain. Useful when a quest object is stuck inside geometry, which does still happen.
player.additem 0000000F 1000 — adds 1,000 gold. The long number is the item's ID, so the same command shape works for anything once you know the ID.
unlock — opens the container or door you have selected. Click the object with the console open to select it first; the command does nothing without a target.
psb — grants every spell in the game, including the ones NPCs use and a few that were never finished.
player.setlevel 50 — sets your level directly.
Two warnings that matter
psb is difficult to undo. It adds hundreds of spells to your book, including unused and debug entries. There is no matching "remove all" command, so the spell list stays cluttered for that character permanently.
player.setlevel does not scale your gear or perks. It raises the level number, which raises what the world spawns against you, without giving you the perks or equipment you would have earned reaching it normally. Jumping too far makes the game harder, not easier.
Achievements
Using the console disables Steam achievements for that session. Mods exist that restore them; whether that is worth doing is your call.
Before you start
Make a manual save first — not a quicksave, which gets overwritten. Console commands are applied instantly and there is no undo.
