1) Closed cell foam mats,like ye olde yellow Karrimat. Typically just under 1cm thick & roll up, though there is also the Thermarest Z-lite that folds concertina fashion, and has an embossed eggbox pattern that's meant to be a bit more comfortable.
Roll size about 50 x 17cm. Tough, don't absorb water, light, fairly good insulation, but not great comfort. These are the lightest, and the cheapest by some long way.
2) Self inflating mats. Ones you'd use for bikepacking are 1 or 1.5 inch thick open cell foam bonded to an airtight cover. When you open the valve the foam expands drawing the air in (top-up by mouth often required). When you lie on it, the foam prevents the rest of the mat bulging, allowing the air to keep you off the ground.
Rolls up to about 27cm x 12-13cm. If the foam is unperforated, warmth is similar to a decent closed cell foam mat, but if the foam has holes punched in it for lightness warmth suffers, so the lighter versions aren't really adequate for winter. Comfort is good if you sleep on your back or front, but if you are a side sleeper your hip can bottom out. Can be punctured (repairable), or the foam and cover can delaminate (not). If the mat delaminates, it's not much better than nothing.
3) Insulated air mats, generally 6-9cm thick.
Uninsulated air beds are cold because of convection. To prevent convection, either an insulator (down or synthetic equivalent) is added, or the air chambers are divided up into small segments by internal baffles. Down insulated mats require pumping rather than blowing to keep the down dry, and occasionally the filter that's meant to keep the down inside when you deflate the mat can leak. It's better to pump synthetically insulated mats too, but it's OK to blow up baffled mats by mouth.
Roll size is between 23x9cm for a smaller baffled mats up to 26x16cm for a larger insulated mat. Comfort is very good due to the thickness. Some people prefer the normal longways tubes over the crossways tubes of the Thermarest baffled mats. Can be punctured (repairable), and occasionally the internal baffles can fail causing bulging (not repairable, but not as much of a problem as delamination on a self inflating mat).
There are very few cheap insulated air mats around - just the Alkpit Nemo than I'm aware of,and this isn't really insulated enough for winter/early sling/late autumn use.
In the specifications, the "R-value" is the indicator of warmth, higher values being warmer*. At a temperature of about zero, and for me using a lightweight down bag & silk liner, a mat with an R-value of 2.5 (thermarest Prolite 4) is marginal, feeling cool, but not so cool I don't get any sleep. YMMV (and it generally does vary). A mat with an R-value of 5 or more will be warm enough for almost all people in all likely conditions.
If you look at Thermarest, don't ignore the Women's versions. They are about 15cm shorter than full length men's versions, but are generally warmer.
Short (120cm) mats are adequate for many people most of the time. They reach from shoulder to knee, the head's on a separate pillow, and the feet don't need very much (empty panniers or a fleece will do).
I've just ordered a Thermarest Xtherm, as the optimum combination of size, weight and warmth
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There are two versions of R-value - American, quoted above, which are measured in BTUs, square feet, hours and °F, and Euro/ISO, measured in Joules, square metres, seconds, and °C. They differ by a factor of 5.7, so R=1 (ISO) is equal to R=5.7 (US). Usually it's fairly obvious which type of R is given.
The XTherm is doing well