In more detail
Internet providers, cloud companies and large organisations have ASNs, such as AS15169 (Google) and AS13335 (Cloudflare). Each autonomous system announces its address ranges (CIDR prefixes) to its neighbours with BGP, the routing protocol that joins the internet's networks together. ASNs are assigned by the regional internet registries (ARIN, RIPE NCC, APNIC, LACNIC and AFRINIC). They were originally 16-bit (1 to 65535) and have been 32-bit since 2007; the ranges 64512–65534 and 4200000000–4294967294 are reserved for private use.
Why it matters
Looking up an IP address's ASN tells you which organisation announces that range. That helps when reporting abuse, telling data-centre traffic from home connections, or blocking a whole hosting provider in a Firewall. The AS holder is the network operator, not necessarily the person using the address.
Example
A lookup result for a documentation address reads like this (AS64496–64511 are reserved for examples):
192.0.2.0/24 AS64500 Example Networks
AS64500 announces the /24, so all 256 addresses from 192.0.2.0 to 192.0.2.255 are reached through that network. Large ASNs are sometimes written in "asdot" form: AS4200000000 is 64086.59904, because 64086 × 65536 + 59904 = 4200000000.