The carry is used for multi-byte arithmetic (amongst other things).
These topics have been written about hundreds of times, so I will just quote a good source.
1.2 REVIEW OF TWOS COMPLEMENT (SIGNED) NUMBERS
There are 256 possible values for a byte; in hex, they are: $00 to
$FF. The range of an 8-bit unsigned number is 0 ($00) to 255 ($FF).
The range of a 16-bit unsigned number is 0 ($0000) to 65535 ($FFFF),
and so on. They are called unsigned numbers because they are zero or
greater, i.e. there is no (minus) sign. A signed number, on the other
hand, can be negative or positive (or zero). The term "signed number"
is used below to mean a twos complement number (although there are
other ways of representing signed numbers). The range of an 8-bit
signed number is -128 to 127. The values -128 through -1 are, in hex,
$80 through $FF, respectively. The values 0 through 127 are, in hex,
$00 through $7F, respectively. So the minimum value of a signed number
is $80 and the maximum value of a signed number is $7F. The range of a
16-bit signed number is -32768 ($8000) to 32767 ($7FFF) ($8000 through
$FFFF are the negative numbers), and so on. This may seem like a
strange way of handling negative numbers, but this method has several
useful properties.
First, 0 to 127 (the overlap of the ranges of 8-bit signed and
unsigned numbers) is, in hex, $00 to $7F, regardless of whether the
number is signed or unsigned.
Second, the most significant bit (bit 7 for an 8-bit number) is zero
when the number is non-negative (0 to 127), and one when the number is
negative. In fact, this is how the N (negative) flag of 6502 got its
name. (Notice that the N flag, when affected by an instruction,
reflects bit 7 of the result of that instruction.) One other note: in
mathematics, zero is not a postive or a negative number, but in the
computer world, things are less formal; the term "positive number"
typically includes zero because (a) all of the other possible values
of a signed number whose most significant bit is zero are positive
numbers, and (b) all of the other possible values for an unsigned
number are positive numbers.
Third, consider the following addition:
CLC
LDA #$FF
ADC #$01
The result (in the accumulator) is $00, and the carry is set. The
addition, in unsigned numbers, is: 255 + 1 = 256 (remember, the carry
is set). The addition, in signed numbers, is -1 + 1 = 0. In other
words, adding (and subtracting) signed numbers is exactly the same as
adding (and subtracting) unsigned numbers.
6502.org also have a detailed discussion about sign and the overflow flag.
