All those answers are way too specific and pointing towards specific technology (e.g. WiFi). The terms repeater, bridge and gateway are much broader than just WiFi.
The similarity between those three devices is that they all "forward" data from one part of the network to another part in the network.
The difference has to do with the OSI model. Repeaters work on layer-1. Bridges work on layer-2. And routers work on layer-3.
A bridge is nowadays most often called "a switch".
Sometimes they are called "layer-2 switches".
Bridges make forwarding decisions based only on the MAC-address (layer-2 address).
A gateway is nowadays always called "a router".
Sometimes a router is called "a layer-3 switch".
Routers make forwarding decisions based only on the IP-address (layer-3 address).
Some devices can do both bridging and routing.
These are often called "layer-3 capable switches".
Repeaters are like amplifiers. They do not look at the content of frames or packets. (A frame is a bunch of data at layer-2. When you look at the same thing from a layer-3 perspective, we call it a packet. On layer-4 it is yet called differently: a segment for TCP and a datagram for UDP).
So repeaters do not look at frames or packets. They just take the incoming signal, bit by bit, and they repeat those bits on different interfaces (an interface is also called a port. It is something you stick a cable in. If you can stick 4 ethernet-cables in a router, we say: the router has 4 interfaces. If the router can do WiFi too, we say that the WiFi is the 5th interface. Etc).
The concept of repeaters is used with many different technologies. E.g. you can have a WiFi repeater, which simply takes the WiFi-signal bit-for-bit, and repeats it. But there are also Ethernet-repeaters. These things are also called "hubs". They just look at the bits, and copy them to all other interfaces. There are also fiber-repeaters. E.g. those fiber-cables under the ocean ? If you send a light-pulse into the cable in London, the signal that comes out in Washington will be way too weak to see. So under the water, the signal gets amplified by repeaters every few hunderd kilometers. There is very little intelligence in a repeater. Just repeat the incoming signal out on all other ports.
A bridge (aka a switch) works on layer-2. Every ethernet interface on every device in the world gets a unique MAC-address. The MAC-address is set by the factory, and never changes. When a computer (or router or laptop or tablet or phone) sends a packet, it puts the MAC-address of the destination in the layer-2 frame (e.g. ethernet frame or WiFi frame). It then sends out the packet. When the receiver with that MAC-address sees that frame, it knows the packet it for itself. So it reads that packet. (Btw, computers and routers learn about MAC-addresses via the ARP-protocol). What a bridge (aka switch) does, it also look at MAC-addresses. And it forwards frame based on the destination MAC-address.
This is called "transparent bridging".
If you want to learn more about it, read this wiki-page:
https://en.wikipedia.org/wiki/Bridging_%28networking%29#Transparent_bridging
A router looks at IP-addresses.
Of course a router deals with layer-2 MAC-addresses too. But only when it talks to computers or other routers that are directly connected. When you communicate over a LAN you use layer-2 MAC-addresses.
But if you want to communicate over the Internet, to computers that are many hops away, then you need IP-routing. Every device on the Internet is assigned a unique IP-address. The IP-address tells us *where* a device is. It is like a postal-code (zip-code). The IP-addresses are advertised and learned via routing-protocols (like OSPF, IS-IS, EIGRP and BGP). Routers talk to each other, using routing-protocols, where all IP-addresses are. They then build a routing-table (also called RIB, routing information base). When a router receives a packet, it looks what the destination IP-address is, it then looks up that address in the routing-table. And there it can see in which direction the packet needs to be forwarded.
- Routing is the most complex technology. But it works with billions of devices in the Internet.
- Bridging is a lot more simpler. But it works only in networks with a few dozen or few hundred devices. If you want to build a bridged network with thousands of devices, you better know exactly what you're doing. Because there are so many complexities and pitfalls, that only professionals can do that.
- Repeaters are the most simple, and the cheapest. But they work only with a few devices. You can't build huge networks with just repeaters.
I hope this is a bit more clear now ?
If you have more questions, don't hesitate to ask.