How do you divide `66,253 div 317`?

(PART 1)

`66253` is the dividend, `317` is the divisor, and the answer is the quotient.

To make things easy and neat, we will construct the long division symbol or the division bracket.
`317sqrt 66253`

First, we will divide `317` by the first three digits of our dividend which is `662`.

317`sqrt` 662`53`

Now, we ask ourselves a question: "how many `317`'s can we get from `662`?". By the process of trial-and-error, we find out that we can get at least 2 `317`'s from `662`. `654` is not "exactly `662`" but it is the closest number to `662`. Multiplying `317` by either `1` or `3` will not give us the closest answer.
`317 * 1 = 317`
317 * 2 = 634
`317 * 3 = 951`

`2` is the first digit of our quotient. Put `2` on top of the dividend and align it on the third digit of the dividend. Alignment of the numbers, especially that of the quotient, is important. (Please don't mind the little dots and dashes; they are not part of the solution.)
................`2`
`317sqrt 66253`

We are not done yet; that was just the first step. Multiply `2` by `317`. The product, which is `634`, will be written underneath `662`.
................`2`
`317sqrt 66253`
...........`634`_ _

(PART 2)

Now, subtract `634` from `662`. The answer is `28`. Write it underneath `634`.
................`2`
`317sqrt 66253`
...........`634`_ _
.............`28`

Oh, and bring down the `5` of the dividend to the difference to get the new number `285`.
................`2`
`317sqrt 662`5`3`
...........`634`_ _
.............`285`

Now, we are back to the first step. How many `317`'s can we get from `285`? The answer is "none" so we instead put `0` next to the `2`.
................`2`0
`317sqrt 66253`
...........`634`_ _
.............285

Repeat the rest of the steps: multiplying `0` by `317`, then subtracting `0` from `285`, then bringing down `3` to the difference. We repeat these steps until "all digits have been processed and no remainder is left".
................`20`
`317sqrt 66253`
...........`634` _
.............`285`
..................`0`
.............2853

For the third time, repeat the rest of the steps: multiplying `9` by `317`, then subtracting `2853` from `2853` to get `0`. We now know that the solution is complete to give us `209` as the quotient.
................`209`
`317sqrt 66253`
...........`634` _
.............`285`
..................`0`
.............`2853`
.............`2853`_
....................0

(It's a long explanation but I hope it helps greatly.)
Sources: https://en.wikipedia.org/wiki/long_division