The Megadrive comes from an arcade machine lineage, it's cut-down Sega arcade machine hardware. The Amiga comes from a computer lineage, it's an evolution of the Atari 800, same designer, and many of the same features, just expanded.
Particularly, the Amiga, like most computers, deals with pixels. Better for things like line drawing, proportional text, and er anything you might want pixels for!
The Megadrive deals in tiles. A tile is an 8x8 block of pixels. They are laid out in a grid of 40x30, as the background. Actually there are 2 background layers, 2 grids, for better effects. Parts of each layer can go on top of, or behind things, according to the game code.
The Master System and NES only had one background layer, so weren't as impressive. You can see that the Megadrive has two, in many games.
Just one access to the tile table in graphics RAM lets you change any tile in the 40x30 grid. 64 pixels, 8x8, can be swapped out in an instant. The Amiga had to do that one at a time, though admittedly it had custom chips to help some things.
For 2D arcade games tiles are a great thing. It's most of the reason why 80s and early 90s arcade machines had such an advantage over home computers, despite using processors that weren't much faster. It's the reason the SNES does so well, even though it's CPU has an 8-bit bus and runs at half the clock rate of the Megadrive's. The graphics chip does basically all the lifting, the CPU just sends it arrangements! By splitting the display into tiles, a processor can throw around graphics at an amazing rate.
So the CPU tells the graphics chip what pixels to put in each tile, often as the level is loading. Then it says which tiles to put in each location of the 2 background layers. Once each tile has been defined, to switch the tiles about is very quick. That's where the power is!
Sprites go on top of the background. Or sometimes behind it, depending on priority! Sprites are made up of a grid of tiles. Each sprite has a table in RAM telling the graphics chip which tiles to use. A sprite can be moved around the screen, rather than staying fixed like the background does.
The Megadrive's multiple layers of screen make parallax effects easy. If you want true parallax, ie per-scanline, that's also a hardware function, a program can add a horizontal offset, a different amount per scanline. Scroll each scanline at a different speed and you have parallax. All for free, done by the graphics chip.
There are lots of interesting differences, but the important one is "tile mapped".
You can do per-pixel access on a Megadrive, by storing your tiles in RAM, and altering a pixel of the relevant tile. If you lay your tiles on the screen in a fairly straightforward order, the formula to calculate what address your pixel is stored at isn't complicated. Still generally slower than a computer though. So there were few polygon games on 16-bit consoles.
is a very good guide to this, with lots of example screens from games with parts of each layer showing or removed, so you can see how it all adds up. Should tell you everything, and more, you need to know! And as a bonus you'll also understand the NES, SNES, Master System, MSX, and most old arcade boards too!
[I've edited this to make it a bit clearer, hope it helps!]