Definition
Every device on a network has a communication address called an IP address so it can send and receive data. An address that an administrator sets in advance, and that stays the same even when the device is turned off and on, is called a "static IP." An address that a service provider or router lends to a device for a set period when it connects, and so can change, is called a "dynamic IP."
An address that must always stay put: the static IP
A server for an online store or a big portal site has to stay in the same place all the time, so customers can find it safely without getting lost.
Imagine a famous department store whose street address changed randomly every morning. Even with a GPS, you would have a really hard time finding it. Exactly the same idea applies in the world of computer networks. Web servers, email servers, file servers, and others must offer their service from the same place no matter when users around the world connect. They run with a fixed static IP that never changes.
A public static IP on the internet is usually a dedicated address that you pay your internet provider to rent. Because the address always stays the same, you don't have to hunt for a new address every time you remote-desktop into your home computer from outside, or check your office security camera live on your phone. That makes it very easy to manage.
A ticket you get fresh each time you enter: the dynamic IP
The smartphones, laptops, and smart TVs we use every day at home, in cafes, and at the office don't need to keep a fixed address at every moment.
Picture a bowling alley with 100 shoe cubbies. If it gave every one of its thousands of members a personal cubby for life, it would run out of space and couldn't operate. But if it lends the key to an empty cubby only to the guests who came to bowl today, and takes it back when they leave, just 100 cubbies can easily serve lots of guests in a single day.
A dynamic IP works on exactly this same efficient idea. Each time a device newly connects to the internet, a router or internet provider picks an IP address that is not being used at the moment and lends it out temporarily. The lent address has a lease period. If the device doesn't renew it and the period ends, the address is taken back and handed out to another device.
A little more precisely: why bother changing addresses?
The most decisive reason for using dynamic IPs is that the number of 4-byte IPv4 addresses that the whole world can share is strictly limited, to about 4.3 billion. If every smartphone, computer, and Internet of Things device on Earth were given a permanent static IP, the addresses would run out in no time.
So, to save this limited resource, networks widely use methods like DHCP (Dynamic Host Configuration Protocol), a protocol that automatically hands out and manages dynamic IPs for devices that connect. A dynamic IP can also change when a device reconnects or its lease ends. That gives an indirect security benefit, because it makes it harder for an outside hacker to learn a specific device's IP in advance and keep attacking it over a long time.
If you use a dynamic IP at home but still want to run a personal blog or a game server, you can solve the problem easily with DDNS (Dynamic Domain Name System), a technology that automatically updates a domain name in real time so it keeps pointing to you even when your address changes often.
Common misconceptions
With a dynamic IP your internet connection is slow, and you need a static IP to be fast.
How an IP is assigned is only a rule about whether the address is fixed or borrowed. It has nothing to do with the speed of your internet line or its data bandwidth.
My computer connected to my home router always has the same address, so it must be a static IP.
Unless you manually fixed it in the router settings, the address may look the same, but in most cases it is a dynamic IP that the router temporarily assigned based on the order devices connected.
Where you meet it
The web servers of portal sites like Google use a fixed static IP so users can reach them any time, 365 days a year.
When you turn your home Wi-Fi router off and on, or switch your phone's airplane mode on and off, the address that gets newly assigned is a dynamic IP.
In one sentence
A static IP is a fixed address that stays the same unless you change it yourself, while a dynamic IP is an address borrowed temporarily each time you connect, to save a limited supply of addresses.