Both IP and network addresses are accompanied by their subnet mask, which is used to define how large or small the network is.
Assigning an IP address (e.g., 192.168.0.1) is a simple as selecting one which is not used within the given network addresses.
The addresses are used for network identification and communication. Unlimited IP address requirements were not considered when the Internet was first conceptualized.
At that time, Internet Protocol version 4, based on 32-bit addressing (IPv4) allowed for 4.2 billion unique addresses.
The protocol used by the router to assign IP addresses is called Dynamic Host Control Protocol (DHCP).
If you have an IP address assigned, it's typically considered a "dynamic IP" because it could be temporary; the router might give the node in question a different IP address at a later time (same with the IP address your ISP gives your router).
Assigning a network address (e.g., 192.168.0.0/24) requires you to understand your current and future needs, so you can use the appropriate network numbering scheme, but also select the best suitable subnet mask.
Being more practical, if you have let's say 155 hosts on your network, you'll surely require a full Class C network address space which provides up to 254 usable IP addresses.
When it comes to your computer(s), there are actually several IP addresses involved.
One is how the computer talks to the internet at large, which is the IP address of your router.