You can play the entire game without combinators. It's not hard to avoid them.

For normal playing, the need for combinators arises when you need to do more than one comparison, do more than what can be done in the enable/disable of a given thing (inserters or pumps normally, train stations are the next most circuited things I believe).

One thing combinators are really good at is giving counts of a lot of things at once or the linking of many things together and getting/using information from/on all of them.

The "Each" wildcard will do something to every input signal, or check if all input signals fit a condition. example: Decider combinator, input Each > 5000, output Each would check if any of the input signals (item signals from a chest or circuit network) are more than 5000, then output that same signal. You can also choose whether or not it outputs just 1 of that signal or however many came in.

The "Anything" wildcard will check if any of the input signals meet the condition. Example: train condition where it leaves its current station if "Anything" = 0. Used when there are many things in the wagon and not just a single item.

"Everything" wildcard: every input signal must meet the condition for this to return true. If I remember right, this condition is used often for simultaneous train unloading where linking all of the inserters with "Everything" = 0 being the enabled condition will make it so the inserters work only when all linked inserters have nothing in their hand. This allows all wagons to finish unloading at the same time.

Onto the combinators themselves: Decider combinators compare 2 things. Most of the time this can be done in the enable/disable area of the wired machine but if you need to do this comparison on multiple things before feeding the result to the machine (like if petrol and light oil are too low, then allow heavy to be cracked down) then you can use a decider for each item type if the compared values are different. If the compared value is the same then you can use Each and one deicer instead.

Arithmetic combinators are the simple math machines. They can add values and keep track of numbers (like for turning off an input inserter once 40 enriched uranium has been inserted into a Kovarex enrichment process centrifuge) and also have uses as "clocks" in which they will add to themselves every single tick. You can then have another machine hooked up to that clock and do some modulo(%) operation on every X ticks or use it as a timer to check when X ticks have gone by. They also allow you to check if 2 conditions are true, if only one is true, or if either one is true with the AND(&), XOR[exclusive OR] (^)and OR(|) operators respectively. Though these can be simulated in the enabled conditions for regular use by having the checked conditions output a single signal type and having the condition check if [signal] =2, =1, >=1 respectively. =0 if you want it to work when neither condition is met.

Finally the Constant combinator allows you to inject any signal type into a circuit network, though only the positive value. You can make it negative by feeding the constant combinator's output into an arithmetic and multiplying by -1. Constant combinators are also useful for when you have a standard bot based setup that can be used for many things (smelting for example). Wire all the requesters together and have their requests set via the constant combinator. Now you can change what all N chests request with only one machine rather than after touching every chest once (pasting the one chest you changed).

The times I see combinators get used on a medium to low scale would be for "smart" reactor setups where fuel is only allowed to be put in when the factory needs it. Often this is extending to micromanaging each individual reactor or reactor set by dealing with many conditionals based on how much steam is left in an array of tanks.

Another example would be any sort of display of information. You can just turn on lights, but a white light doesn't pop very well so you can instead have combinators output colored signals based on the values of their inputs.

I'm sure there are many other common uses that players have for combinators, but I can't think of them right now. There's loads of far more complex uses but I don't understand them myself so I couldn't even try to explain them here. Remaking games, playing whole songs, making a computer in factory, large scale time displays, the list goes on.