If you have played many games I'm sure you know what Mods are. If you don't, then they are Modifications to the game usually made by the community. They are like add-ons of sorts that change or enhance the gameplay in some way.

Factorio has a very large and active Modding community so if you are looking for some ways to change how the game plays you will probably be able to find a Mod for it. The Factorio Developers seem very supportive of the Modding community as well which is great.

You can check out the mod portal website here, where you can upload a mod, browse existing mods, rate and comment on mods. Uploading a mod requires you to pick one of the available licenses, or to specify your own.
As with the multiplayer matching server, using the in game mod portal requires you to login with your Factorio website account, and to download mods from the website also requires a login.

There are multiple different categories of Mods with some of them just adding a few additional items to the game, or some changing a few aspects of how the game actually plays, and even some that almost overhaul the whole game and crafting system. Unless it is something that is impossible to Mod or very strange, you will most likely be able to find a Mod to do what you want. I will go over a few Mods that each change something in a different way and are best for certain things. However before we move on I would like to say that I highly suggest you play and get a grasp of the vanilla game first before you do anything with Mods.

If you would like to watch video spotlights on some of these Mods as well as many others you can check out this forum thread where they are listed.

If you want some Mods that will just enhance specific parts of the game and not revamp any recipes or actual gameplay than I would suggest some of the following...

Fully Automatic Rail Layer (FARL): This Mod essentially lays rails automatically if you just drive forward. Makes putting down long sections of rail much easier. More info can be found on the forum section for it linked above.

Advanced Logistics System: Keeps track of every single item in the Logistics System and allows you to view exactly what is in each chest and much more. Essentially makes it a lot easier to keep track of things in the system.

Initial Map Scan: When you first start a new map this Mod scans and reveals (removes fog of war) an area the size of the starting area you selected in the map options. Basically makes it so you don't have to walk around and explore the area yourself before deciding to move on or not.

Obviously these are not the only mods that change small parts of the game, and I'm not saying they are better than any of the others. I listed these because I have personally played with them and know a lot of other players that have as well.. This applies to all the mods I list here. I would definitely recommend checking out all Mods on the forums that seem interesting.

Now if you want some Mods that change how the game plays a little bit but don't really overhaul it then something like these maybe of interested to you:

Resource Spawner Overhaul: Completely changes how the resources spawn on the map. Creates more of a need for trains because it makes the resources spawn farther apart and also makes them fairly scarce in the starting area but increased richness the farther out you get. Spreads out Biter bases as well.

Treefarm Lite: Does exactly what the name says and adds farm to grow trees in. So essentially gives unlimited wood that you don't have to manually chop down because it can be fully automated. This mod is best used in conjunction with some others like Dytech or Bobs Mods which will talked about a little later.

Again, there are many more mods like these that you shouldn't neglect. These three are just some that I have personally used and quite enjoyed.

Lastly, if you want something that completely overhauls the game, changing recipes, adding tons of new items and technologies, and even new enemies then I would suggest these:

Note: You should have a firm grasp on the base game before you try these types of Mods. They are quite complicated and can be confusing even to more experienced players.

Dytech: Dyetch is almost a complete game overhaul. It leaves all the vanilla items in the game and they still have uses but it changes lots of the vanilla recipes and adds tons of new items and extra levels of existing items. It is modular which means you can kind of pick and choose which parts you want to use and which you don't. One of the modules is specifically for the enemies and makes the game much, much more difficult by adding a lot of new enemies. Turning this part on is not for novice players as you will get crushed very quickly.

Bob's Mods: Similar to Dytech in the fact that it overhauls the game. Bob's Mods is more complicated and complex. It takes perhaps a more realistic approach to the item recipes and progression which makes things more interesting. Also changes the enemies and adds new ones. Bob's is modular just like Dytech so you can pick and choose which parts you want.

I hope these suggestions can get you started and give a good idea of what type of things the Mods for Factorio can do.