Patterns are shortcuts: combinations of numbers along a wall of hidden tiles that always resolve the same way. In the diagrams, a red ! is a certain mine and a green ✓ is a certain safe tile.
1-1 along a wall
Two 1s side by side along the edge of the hidden area, starting from a wall: the first 1 only sees two hidden tiles, and both of those are also next to the second 1. So the second 1's mine is one of those two, and the next tile along is safe.
1-2
A 1 next to a 2 along a straight edge: of the tiles the 2 sees, the ones it shares with the 1 can hold at most one mine. The 2 needs two, so the tile on the far side of the 2 is a mine.
1-2-1
One of the most common shapes. The two outer tiles above the 1s are mines, the one above the 2 is safe.
1-2-2-1
The longer cousin. The two middle tiles above the 2s are mines, the outer ones above the 1s are safe.
The corner 1
A 1 with only one hidden neighbour, often in a corner of the board: that neighbour is the mine, and finding it usually unlocks the edge next to it.
Reduce, then repeat
The real trick is reduction: subtract the mines you already know about from each number. A 3 that already touches two flags behaves like a 1. Once you reduce the numbers, the same small set of patterns shows up again and again.
Practise them
Patterns stick through repetition. Run a few Intermediate and Expert boards, try the Daily board, or use them against real people in multiplayer minesweeper, where spotting a 1-2-1 half a second faster wins you the tiles.