From Neoxis' Guide which is based on Meow's Guide with help from Intheisto.

Windows user:
C:/ users/ yourname/ appdata/ local/ nuclearthrone/
(Make sure you can view hidden folders through your view tab in the control panel's folder options)
XP:
C:/ Documents and Settings/ yourname/ Local Settings/ Application Data/ nuclearthrone

Mac user:
Users/ yourname/ Library/ Application\ Support/com.vlambeer.nuclearthrone
"Also, to get to the Library folder on Mac you have to click on the "Go" tab on the very top of the screen and hold "option" for it to appear."

Linux user:
*home folder*/.config/nuclearthrone

Once in the folder with the game save files create a new text file labled "seed".

After the text file named "seed" is created edit the text in the file and save it after it is edited. Only numbers will effect the seed. Anything that is not a number, including letters and blank spaces, will be counted as a zero by the game when selecting the seed. The character limit for numbers in a seed file is 19; inserting more than 19 characters will result in the same seed as if no characters had been entered (there are 9223372036854775296 unique seeds).

As of update 74 seeds are broken. The game will not crash because of the use of seeds, however seeds will often fail with a variety of glitches including, but not limited to; terrarian deformation, disco parties, interchanged level selection, mass confusion, unintended enemy ratios, cave foreclosure, pre-opened chests, highly contagious uncontrollable bowels, unintended spawn locations of everything, death, chest progress resets, and the skittley-hipps.

All credit for Nuclear Throne goes to Vlambeer and associated persons.

Information related to seeds is courtesy of JW and stored on the Official Nuclear Throne Wiki.
Who is Gieron?

Thanks to Meow and Neoxis for their work on previous tangentially related guides.

Solid is a derp.

Mutations are currently fixed to seeds without exception. When Crown of Destiny is used the seed will still have consistent mutations. In the event that a mutation is taken when that mutation would normally appear later a predetermined mutation will take it's slot in on the selection screen.

Under normal circumstances (no crown) seeds are mostly stable. There is always a chance for the game to generate a level based on a secondary layout on each level generation, however this chance is low and will always generate the same secondary level layout when it happens.

Future level layouts can be modified by visiting secret levels and vaults. After visiting a non-standard level future levels will have terrain deformation near spawn and modified enemy ratios; these effects decrease as the player progresses through future levels.

Chests are normally predetermined including large weapons chests. Large weapon chest spawns will happen based on the seed; each time a player skips a weapon chest the seed will determine which future level will have the large weapon chest. Each time a weapons chest is skipped after the first the seed will select a new level for the large weapons chest that is equal to or closer than the previous level selected; all of this is predetermined.

All weapon drops are predetermined, however the drops will be modified base difficulty, Robot's passive and cursed chests; these changes will be predetermined based on the seed as well.

Seeds with crowns are entirely broken with the exception of Mutations. The use of a crown will cause level selection, chest spawn, enemy variety, and weapon spawn to be RNG based. There are a selection of predetermined levels a crown run will select between, however there is no way to predict what will come next.