The board is central. Literally and figuratively.
The board sits between the players. You'll keep your cards on your side of the board. The dealer keeps their crib on their side. No touching.
Points accrue quickly during play, and a board is easier (in the long run) than a pad of paper and a pencil. There are probably phone apps that mimic a board. A simple block of wood with rows of holes and metal pegs is ideal. Wooden pegs are traditional. Plastic pegs are slippery nonsense.
Pegging
The pegs "leapfrog" down the board. The forward peg shows the current score. The back peg is the previous score.
The back peg is the only one to advance; it jumps past the forward peg.
You can check to make sure the other player counted the holes properly. It's a handy visual audit of scoring.
This peg movement is pretty easy -- most of the time. Once in a while, you'll mess up and grab the wrong peg.
The Board
There are two common varieties of boards: "continuous" and "track" boards.
Beginners often find the continuous boards easier to use. There will be 121 holes in a serpentine or spiral pattern. The first one to get past 120 points is the winner.
Let's say you're in hole 120 -- called the stink hole -- where any single point make you the winner. You may be holding a "pair royal" that gets you six points. You drop the peg into the magical extra hole and enjoy your moment of victory. A few extra points don't matter; this isn't darts where you have to go out with the right number of points. Cribbage is "first over the line".
If you're very competitive, you'll track two aspects of your victory:
- You won.
- How many points behind you the other player was. This can be handy for adjudicating the awkward situation where you play an even number of hands and the win/loss score is tied.
All four pegs started together. Only one pag cross the line at the end.
Track Boards
I grew up with track boards. On this style board, each player will use a pair of parallel tracks, 30 holes in length. A game, then, is four trips across the board.
There's also one extra line of holes. All four pegs start here. When someone wins, their peg will finish here.
Many boards will have two groups of ten or so extra holes. These are for counting the number of games won. The board my partner's dad made for us doesn't have a few extra scoring holes: it has three tracks; we use the middle track to record games won. Up to 30, since that's the number of holes. We then transfer the count to a spreadsheet. Really.
With a track board, you'll go out on one track, turn the corner can come back on the other. This will be done twice. The first 30 points are first street. Then, you turn the corner, and the next 30 points will go back down toward where you started on the other track. This is called second street.
Counting holes around the corner is tricky for me. I'm not a visual thinker. My partner, however, can easily see three holes on third street and nine holes around the corner on fourth street; they flawlessly drop the peg in the right spot. I have to count down to the end, and then back to be sure they did it correctly.
After turning a corner, it's sometimes a little hard to recall which peg is front and which peg is back. Like our hand-made board, track boards are often blank. Since pegs are going up one of the two tracks and down the other, it's common to grab the wrong peg and move in the wrong direction.
The worst is when you're over 30 points behind the other player. They'll be going one way on -- say -- third street. You'll be going the opposite way on second street. Try to be mindful of which direction you're going even when it looks bleak. Changes of fortune are common; the lead can move back and forth.
My grandfather's rule for using a track board was "up the outside, down the inside". This was pleasant because it didn't matter how the board was oriented. One player is moving their pegs clockwise around their two tracks, the other player is going counter-clockwise.
Here's the board we use:

My partner and I always place the board with the extra starting and ending hole on my left (their right). We've been doing this way for years. We start down first street with me moving pegs to my right and them moving pegs to their left. The picture is from my vantage point.
We go follow the "Port Out Starboard Home" (POSH) rule. We consider the starting/ending hole (on the left end of the board from my view) to be the "stern" of the board. We head out on the left-side track from the board's perspective. We head back on right-side track. The idea of the board-as-a-boat with a defined port and starboard comes from actually living on a boat.
(In the boat, if we played in the saloon, the board was actually athwartships. The picture is from the cockpit. The board was oriented the way the boat points; the bow is to the right in the picture.)
Continuous boards are generally better for beginners. One track. One direction. No Up the Outside, Down the Inside rule.
You can't have too many boards. When you have friends over to visit you can have two games and switch players to make sure everyone plays with everyone else.
Next, we'll overview the game. It's played in phases, with a metric ton of complicated-seeming scoring that happens throughout.