One address, several spellings
An IPv4 address is a 32-bit number. Dotted decimal notation is only a convenience: 192.168.1.1 is four 8-bit values written in base 10, separated by dots. The same quantity can be expressed as a single decimal integer, 3232235777, as hexadecimal 0xC0A80101, as binary 11000000.10101000.00000001.00000001, or as octal 0300.0250.0001.0001. Every one of those strings decodes to the identical 32 bits, and network equipment will treat them as the same address.
The converter takes any of these forms and prints the rest, so a value lifted from a database, a packet capture or a firewall rule can be read in the notation you actually think in.
The arithmetic behind the conversion
Each octet is a base-256 digit. The first octet is multiplied by 16,777,216, the second by 65,536, the third by 256, and the fourth is added as is. Sum the four and you have the integer form. The largest possible value is 4,294,967,295, which corresponds to 255.255.255.255. Going the other way, divide by 16,777,216 and take the quotient and remainder, then repeat with 65,536 and 256.
Two practical warnings. Values above 2,147,483,647 do not fit in a signed 32-bit integer, and 192.168.1.1 is one of them, so a column typed as a signed INT will overflow on most home router addresses; use an unsigned 32-bit or 64-bit column. And leading zeros are ambiguous: some C library parsers, including the classic inet_aton, read 010 as octal 8, so 192.168.010.001 does not mean what it looks like.
Converting 192.168.1.1 both ways
Convert 192.168.1.1 by hand. The first octet contributes 192 x 16,777,216 = 3,221,225,472. The second adds 168 x 65,536 = 11,010,048, taking the running total to 3,232,235,520. The third adds 1 x 256 = 256 for 3,232,235,776, and the fourth adds 1 for 3,232,235,777. In hex, 192 is C0, 168 is A8, 1 is 01 and 1 is 01, so the hex form is 0xC0A80101, and in binary the same bytes are 11000000 10101000 00000001 00000001.
Run it backwards to check: 3,232,235,777 divided by 16,777,216 is 192 with a remainder of 11,010,305. That remainder divided by 65,536 is 168 with 257 left. 257 divided by 256 is 1 with 1 left, giving 192.168.1.1 and confirming the round trip.
Where each notation shows up
Integer storage is common in geolocation databases and access-control tables, because a single indexed integer column answers "which range contains this address" faster than string comparison, and range queries become simple inequalities between two numbers. Hex appears in packet captures, firewall logs and any place where an address is packed into a fixed-width field. Binary is the form to think in when reasoning about subnet masks, since a mask is just a run of ones followed by a run of zeros.
Once you have the integer, the next questions are usually about the mask and the block. CIDR to range expands a prefix into first and last address, and IP class finder reports the historic class and whether the address sits in a private range. For the public address of the machine you are on, IP lookup reports what the internet sees.
