Reading an address
IPv4 addresses are 32 bits, about 4.3 billion in all, and they have run out. IPv6 addresses are 128 bits long. That gives about 3.4 × 10^38 addresses, enough for every device to have its own.
The 128 bits are written as 8 groups of 4 hex digits, with colons between them. Hex digits run 0–9 then a–f, and each stands for 4 bits. So 0db8 is 0000 1101 1011 1000: 16 bits per group.
That is long, so there are rules to shorten it (RFC 5952). First, drop the zeros at the front of each group: 0db8 becomes db8, 0001 becomes 1, and 0000 becomes just 0. Letters are written in lowercase.
Next, replace the longest run of all-zero groups with two colons, ::. Do it only once, or nobody could tell how many zeros each gap hides. The router's address shrinks to 2001:db8:1::1.
Parts and kinds
On a LAN the first 64 bits are the network prefix, written /64: here 2001:db8:1::/64. The last 64 bits are the interface ID, which names one device. One LAN has room for 2^64 interface IDs.
Addresses that work across the whole internet are global unicast: one sender, one receiver. They come from 2000::/3, so they start with 2 or 3. The block 2001:db8::/32 is set aside for examples like these.
SLAAC
When your laptop joins, it first gives itself a link-local address: fe80:: plus an interface ID, here built from its MAC address. Every IPv6 interface has one. It works only on this link; routers never forward it.
IPv6 has no broadcast. To reach a group it uses multicast addresses, which start with ff. Your laptop sends a Router Solicitation to ff02::2, the group only routers join: is there a router here?
The router answers with a Router Advertisement to ff02::1, the all-nodes group, so every device hears it. Inside is the prefix card, 2001:db8:1::/64, with a flag that says devices may build their own addresses.
Your laptop puts the prefix in front of its interface ID: 2001:db8:1:0:200:5eff:fe00:5310. It checks first that no one else uses it (Duplicate Address Detection). No server hands it out: this is SLAAC, stateless address autoconfiguration.
An ID built from the MAC stays the same on every network, so websites could follow you. So the laptop also makes temporary addresses with a random interface ID for outgoing connections, and a fresh one about once a day (RFC 8981).
The address, worked out
IPv4 2^32 = 4,294,967,296
IPv6 2^128 = 340,282,366,920,938,463,463,374,607,431,768,211,456
≈ 3.4 × 10^38
example: the router's address 2001:0db8:0001:0000:0000:0000:0000:0001
All global addresses here are inside 2001:db8::/32, reserved for documentation (RFC 3849); the MAC addresses 00:00:5e:00:53:10 and 00:00:5e:00:53:20 come from the RFC 7042 documentation block. The temporary interface ID is an example of a random value. Building the interface ID from the MAC (modified EUI-64, RFC 4291 Appendix A) is the classic method; many systems now use a stable random-looking ID instead (RFC 7217, recommended by RFC 8064).