Bitwise Operations exist in programming languages as a direct way to modify the binary values, or bits, in a number. Sometimes, it's more efficient or useful to modify the bits of a number rather than performing arithmetic on the number. One example of this is a bitmask where you use the BAND operation to choose several bits to check and ignore the others.
Pico8 has a collection of useful bitwise operations. Keep in mind that, since update 0.2.0 when they were added, the operator forms of the bitwise functions run faster in the simulated CPU than the API Calls do.
Operation Name  API Call  Operator  Description 

Bitwise Not  bnot(a)

~a

Inverts all of the bits of the number. 0 swaps to 1 and 1 swaps to 0. (Invert) 
Bitwise And  band(a,b)

a & b

Compares the numbers at each bit position. If the bits are both 1, the result is 1, otherwise if a 1 is compared to a 0, the 0 replaces it.. (Zerodominant) 
Bitwise Or  bor(a,b)

a  b

Compares the numbers at each bit position. If the bits are both 0, the result is 0, but if a 0 is compared to a 1, the 1 replaces it. (Onedominant) 
Bitwise Exclusive Or  bxor(a,b)

a ^^ b

Compares the numbers at each bit position. If the bits are the same, the result for that position is a 0, but if a 0 is compared to a 1, the bit is "Exclusively" in only one input, and the result is a 1. (There can be Exclusively One) 
Shift Left  shl(a,b)

a << b

Shifts each bit of a number left by b positions. 
Arithmetic Shift Right  shr(a,b)

a >> b

Shifts each bit of a number right by b positions. Arithmetic shift attempts to preserve the sign (Positive or negative) of the number by copying the sign bit (farthest left) into the resulting empty positions. 
Logical Shift Right  lshr(a,b)

a >>> b

Shifts each bit of a number right by b positions. Logical shift ignores the sign of a number, filling zeroes into the space left over by the shift. 
Rotate Left  rotl(a,b)

a <<> b

Rotates each bit of a number left by b positions. In a rotation, bits which "fall off" on the lefthand side are wrapped around to the right of the number. 
Rotate Right  rotr(a,b)

a >>< b

Rotates each bit of a number right by b positions. In the rotation, bits which "fall off" on the righthand side are wrapped around to the left of the number. 