This is one of the basic tools I use for cooking. I've explained how to use it before how to make it, because I think it's easier to understand that way.
Using white sauce
The basic sauce with no modification can be used as a sauce for vegetables or pasta. It's fairly bland unless you add herbs or spices for flavour, however.
Melt cheese into the sauce (at any stage, probably best done after the sauce itself is made if you're a beginner) to make a cheese sauce for pasta, vegetables and some meats.
Add frozen vegetables and let the sauce heat them, or add tinned or cooked vegetables after the sauce is made, to make a vegetable mornay. Tinned asparagus works really well. You can substitute the liquid from tinned vegetables for some of the milk to add more flavour.
Add cooked white meats to a vegetable mornay to make a chicken or fish mornay. Very yummy. You can substitute stock or dripping (the liquid from cooking the meat) for some of the milk to add more flavour.
(If you use dripping, look to see if it has two layers. If it does, the top and comparitively clear layer is fat. Skim that off and use the lower layer only.)
To make lasagne, make a white sauce with cheese melted into it, and make a saucepan of your choice of meat (I cook fat-trimmed mince with a can of chopped tomatoes and a bunch of mixed veg). Layer the lasagne with lasagne pasta at the bottom, then repeat a pattern of mince mix, lasagne pasta, cheesey sauce and lasagne pasta, topping with cheesey sauce. Bake as directed on the lasagne packet.
To make a gravy instead
Substitute the milk for stock (or dripping, see above) and water.
For 'white' gravy, use stock and milk.
For vegetable gravy, use vegetable stock (purchased or home-made).
(To make your own veggie stock, use the water your veggies have been boiled or steamed with. Or save the offcuts of your veggies - broccoli steams and celery tops and carrot tops and the like, and boil them. Keep the water, throw the solid stuff into the garden as mulch. That water is your stock.)
Making white sauce
Approx 1 tablespoon olive oil (or your preferred alternative)
Approx half a cup of flour (or gluten-free flour or anything like that)
Approx 1 litre milk (or soy milk or stock or water or almost any water-based liquid)
Salt, pepper, herbs and spices to taste
Into a large saucepan, put the olive oil and the flour. Mix them together with a wooden spoon (or other stirring tool) until they're fully mixed, then put the saucepan on a hotplate at medium heat.
Add a small amount of milk and stir it in. Keep adding milk and stirring for a while - see below.
Here's the tricky part: you've got a chemical reaction going, and you want to control that reaction. You do it by controlling how much heat you apply. I recently realised that what I do is hold the saucepan with my off hand and stir with my right, and I keep moving the saucepan to get a fine level of heat control.
Watch the mixture when you add milk and stir - you'll see that there's some areas where the milk is truly blended in with the thickening sauce, and some where it isn't. You want to get the milk all blended in, which is part of why you stir.
You'll also notice that where there's more heat, the sauce is thickening faster. This is the other reason you stir - to try to get it to thicken evenly.
If you find you get it lumpy, take it off the heat and stir it smooth, then try adding a little more milk before you put it back on the heat. Turn the heat down, and make the sauce with lower heat until you get better at it.
As you get good at making this, you can use higher heats. If you're not so good yet, you can slow it by using lower heat.
Continue adding milk and stirring it in until the sauce is at a thickness you like, and if it hasn't boiled at that point, turn the heat up just a bit and keep stirring (and adding liquid to maintain the right thickness) until it does.
Add the salt, pepper, herbs and spices at any stage. As a beginner, add them when the sauce is almost ready.
As an expert, put herbs and spices in the oil and heat the oil before you add the flour. This brings out a bit more of the flavour of the herbs and spices, but can add a bit to the challenge of keeping the sauce smooth.