The difference between a static and a dynamic IP address is that the first (static) never changes and always stays the same, while the second (dynamic) is always changing.
Typically, IP addresses assigned by ISPs for home networks are dynamic because they are easier to manage and deploy than static. On the other hand, static IP addresses are more often used by business networks.
When it comes to the advantages, static IP addresses include better DNS support, are easier to use with VoIP and better work with remote access programs like VPN.
On the other hand, dynamic IP addresses are typically cheaper and easier to configure and practically have unlimited IP addressing since they reuse IP addresses. In addition, they also make hacking somewhat harder as it's not that easy for a hacker to find out where you are.
When it comes to the disadvantages, static addresses cost more and allow the hacker to know exactly where the IP owner is. Dynamic addresses don't work that well with geolocation services, they can suffer from downtime more often and may not work well with VPNs and other remote access programs.