IPv4
In short: The fourth version of the Internet Protocol — 32-bit addresses, classically written as four numbers separated by dots (e.g. 192.168.0.1), around 4.3 billion possible addresses.
In more detail: Because 4.3 billion addresses have long ceased to be enough for the number of devices connected today, IPv6 was developed. However, because of its enormous spread and compatibility, IPv4 remains the de facto standard, often running in parallel with IPv6 (“dual stack”).
In Depth
An IPv4 address consists of 4 bytes (32 bits), usually written in “dotted decimal notation” — four numbers from 0 to 255 separated by dots, where each number represents exactly one byte:
192 . 168 . 1 . 42
11000000.10101000.00000001.00101010 <- the same address in binary
Mathematically, 32 bits give 2^32 ≈ 4.3 billion possible addresses — a number that seemed enormous in the 1980s, but is nowhere near enough today given smartphones, IoT devices and billions of connected users. The last freely available IPv4 address blocks were handed out to the regional registries years ago (“IPv4 exhaustion”).
In practice this shortage is cushioned mainly by two techniques: private IP address ranges (every household/company uses the same private addresses internally, without needing global uniqueness) combined with network address translation (NAT), which hides many private addresses behind a single public address. This works well enough that the switch to IPv6 — despite its technical superiority — has been progressing only sluggishly for years: many networks run in “dual stack” mode, supporting IPv4 and IPv6 in parallel, because completely replacing IPv4 is practically impossible to enforce.
See also: IPv6, IP addresses, Subnet mask