an autonomous system number
If every independent network on the Internet is a town, an autonomous system number (ASN) is that town's unique postcode. It is a globally unique integer that names one autonomous system, so that when networks talk about routes they can say exactly whose network a path passes through. Without these numbers there would be no way to write down or reason about Internet-wide routes.
ASNs are handed out by the same regional registries (RIRs) that manage IP addresses — ARIN, RIPE, APNIC, and so on — so that no two networks ever share a number. Originally an ASN was a 16-bit number, giving 65,536 possibilities (about 2^16). The Internet outgrew that, so the format was extended to 32 bits (about 2^32, over four billion). A 16-bit ASN like AS3356 simply looks like a small number; 32-bit ASNs are often written in plain decimal too, such as AS400644. In BGP, an AS path is literally a list of these numbers showing which networks a route crossed.
A useful caveat: an ASN identifies a routing entity, not the value or trustworthiness of a network. A brand-new AS with a high-numbered ASN is no less legitimate than an old low-numbered one. And a single ASN can carry an enormous amount of the world's traffic (a tier-1 backbone) or almost none (a small company). The number is just a name.
Cloudflare uses AS13335, Hurricane Electric uses AS6939, and Comcast uses AS7922. When you run a traceroute across the Internet and a tool annotates each hop with "[AS13335]", it is mapping the router's IP address back to the ASN that owns it.
ASNs let humans and routers name exactly which network owns an address or sits on a path.
Some ASNs (the 64512–65534 and 4200000000+ ranges) are reserved as private and must never appear on the public Internet — like private IP addresses, they are for internal use only.