Since tcp usages are pretty straightforward from other answers, I'll mention some interesting UDP use-cases:
1)DHCP - Dynamic Host Configuration Protocol, which is being used in order to dynamically assign IP address and some other network configuration to the connecting devices. In simple words, this protocol allows you just connect to the network cable(or wifi) and start using the internet, without any additional configurations. DHCP uses UDP protocol. Since the settings request message is being broadcasted from the host and there is no way to establish a TCP connection with DHCP server(you don't know it's address) it's impossible to use TCP instead.
2)Traceroute - well-known network diagnostic tool which allows you to explore which path in the network your datagram passes to reach it's destination(and how much time it takes). By default, it works by sending UDP datagram with unlikely destination port number(ranging from 33434 to 33534) to the destination with the ttl(time-to-live) field set to 1. When the router somewhere in the network gets such datagram - it finds out that the datagram is expired. Then, the router drops the datagram and sends to the origin of the datagram an ICMP(Internet Control Message Protocol) error message indicating that the datagram's ttl was expired and containing router's name and IP address. Each time the host sends datagrams with higher and higher TTL, thus increasing the network part which it succeeds to overcome and getting new ICMP messages from new routers. When it eventually reaches it's destination(datagrams TTL is big enough to allow it),- the destination host sends 'Destination port unreachable' ICMP message to the origin host. This way, Traceroute knows that the destination was reached. Since the TCP guarantees segments delivery it would be at least inefficient to use it instead of UDP which, in turn, allows datagram to be just dropped without any resend attempts(resend is implemented on the higher level, with continuously increasing TTL as described above).