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Notes -
DHCP requests are transmitted over UDP with a target destination of the broadcast address, usually 255.255.255.255. The standard says that this packet should have a source address of 0.0.0.0, but in my experience most DHCP servers aren't very picky about that. This packet is just a message going across a wire to every receiver on the local network (ie, up until the gateway), so the ethernet card doesn't need to have an IP address at that time. EDIT: for clarity, it uses the MAC address to identify itself and so the server can properly respond to just the correct machine. This is one of many reasons that getting DHCP to run across network boundaries is an absolute nightmare. /EDIT
For the purposes of TCP/IP, the local network is defined by the netmask. Physical networks (eg, having multiple routers with different subnets plugged into the same big switch) and logical networks (VLANs) can and often are different. This is a space with a lot of namespace collision, so be wary of it.
At the risk of going too deep into the (lies-to-children!) OSI model:
Before doing anything else, the sending computer looks at its ARP table, which converts IP addresses to MAC addresses. If the destination IP address is not on the ARP table, it will send an ARP request, which is a broadcast message to the local network asking if any devices have that IP address (or, if not on the local network, it sends an ARP request for the local gateway). Once it finds the address, it inserts that IP-MAC pair into the ARP table, and uses it as part of the packet and frame shaping.
The computer forms a packet, with a source IP address of 192.168.1.2/24 and a destination of 192.168.1.3/24, at the TCP/IP network layer, or layer three. The ethernet card breaks this into one or more "frames" with a maximum size called the MTU (historically 1500 bytes, but can be larger where hardware supports it), aka the ethernet/MAC data link layer or layer two. It then transmits these frames as signals to the network switch, aka the ethernet physical layer or layer one.
This switch will receive the signals, and convert them into the layer two frame. On older hubs, it would simply echo the frame out every port. On modern switches, it then inspects the frame for a destination MAC address. If the switch has records of receiving frames with a source MAC address matching that destination, it only sends the frame to that specific physical port or ports. If it has no record, it floods the frame out every port, and it's up to the receiving device to filter whether it's address properly. But the switch tables get filed with records pretty quickly
((For older computers, there was a physical layer conversion issue; this is why crossover cables existed. But almost every modern device can automatically switch over.))
In that case, the frame would be configured with a destination MAC of the local gateway, so the switch would look in its MAC table for the MAC of the local gateway, and usually only send the packet to the physical ports of the local gateway. This is layer two switching, not layer three routing.
It's only when the frame gets to the gateway, which reassembles the frame into a packet to inspect the destination IP address, that the gateway examines what the target IP address is, and then routes it by checking its own routing tables and own default gateway.
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