Joining the LAN
Your laptop just joined the home network. It has a MAC address (a fixed hardware name) but no IP address yet, so it cannot talk to anyone. DHCP, the Dynamic Host Configuration Protocol, will lend it one.
DORA
The laptop shouts to the whole network: a broadcast from 0.0.0.0, port 68, to 255.255.255.255, port 67. Every device hears it, the phone too. Only a DHCP server answers.
The router is the DHCP server here. It looks in its pool, 192.168.1.100 to 192.168.1.199. The phone already holds 192.168.1.100, so it sets aside 192.168.1.101 for the laptop.
The router replies with an offer: the address 192.168.1.101, a lease of 24 hours, and its own address as the server ID. A lease is a loan: the address is yours only for that long.
The offer carries settings, called options: the subnet mask 255.255.255.0, the router 192.168.1.1 as default gateway (the way off the LAN), and a DNS server for looking up names. The laptop has not said yes yet.
The laptop asks for 192.168.1.101 from server 192.168.1.1. It still broadcasts, because it has no address yet, and so that any other DHCP server that made an offer learns it was not chosen.
The router writes the loan in its lease table, 192.168.1.101 for MAC 00:00:5e:00:53:01, and confirms with an ACK (acknowledgement). Those four messages, Discover, Offer, Request, Ack, spell DORA.
The lease
The laptop takes the address 192.168.1.101/24 and the gateway and DNS settings. Now it can reach the internet. Its lease clock starts counting the 24 hours.
At T1, half the lease (12 h), the laptop asks to keep its address. Now it has an address, so it sends the DHCPREQUEST straight to the router (unicast), not to everyone.
The router says yes with another DHCPACK. The laptop keeps 192.168.1.101, and its lease clock starts again from zero, so a device that stays connected keeps the same address.
Had no answer come, the laptop keeps trying until T2, 87.5% of the lease (21 h). Then it broadcasts its request to any DHCP server. If the full 24 hours pass, it must stop using the address and start over with a Discover.
The message, field by field
;; step 1/11 · no-address · state ;; at the laptop interface wlan0 MAC 00:00:5e:00:53:01 IPv4 none (0.0.0.0) gateway none DNS none DHCP state INIT
A 24 h lease is a common home-router default; T1 = 50% and T2 = 87.5% of the lease are the RFC 2131 defaults, and a server may send other values in options 58 and 59. Before using a new address, the client should check nobody else has it, for example with an ARP probe; if someone answers, it sends DHCPDECLINE and starts again.