I have always used the actual weight of a yarn to represent the thickness, usually written as meters or yards per skein or 50 g, 100 g or 200 g. But it is not always the best way as different materials have different density. So, if you want to combine different materials in one project, you need to make sure the thickness of these yarns is about the same.
The best way is to actually take a look at the yarns, but it is not always an option. Another way to compare yarns is to look at the suggested hook or needle sizes. You can also look at the gauge which is usually given as stitches per 10 cm (4 in) knitted in stockinette stitch.
There are also many more ways to represent the yarn weight, which, unfortunately, we do not use in Estonia. But here is a chart that might help you find the right yarn for any project. Click on the chart for a bigger view.