I have seen two words for "car" in Russian: "машина" and "автомобиль" or "авто". Which is most common?
The word most common in speech is машина.
Автомобиль is the "right word", but it's long and unpopular. Авто (unlike, say, its German counterpart) is not a "real word" on its own, but rather a spoken short form of автомобиль, or just a prefix. It has its usage (say, in magazines), but generally it's far behind машина.
There are also slanguish variants, of course. The most popular are тачка (and its derivatives тачила, точилка etc.) (cf. "ride" in "Pimp My Ride") and колёса (cf. "wheels").
BTW. In XIX century speech машина mostly denoted "train". I bet, in the future when we won't have "cars" anymore, the word машина will still be in use, meaning some kind of small spaceship and such.
I would say that "машина" is the only correct version. "автомобиль" is completely dated and today is only used for affectation or humorous effect, while "тачка" and "мотор" is pure slang and could never be used formally.
Lots of terms related to cars though are formed with a prefixoid АВТО - such as "автостоянка" (car park); "автозаправка" (petrol station); "автосервис" (car repair) etc., etc. Note (in it's turn dated, but widely used in the 60s-80s) "автолюбитель" - literally "car lover" or "car connoisseur", which meant simply a car owner - at a time, when car owners were few and far between.
Don't forget that "машина" also translates as "machine", such as "стиральная машина" (washing machine), "посудомоечная машина" (dish washer) and so forth, as well as "машина времени" (time machine (as in science fiction); "государственная машина" (machinery of government), ect.
Hope it helps.
official: автомобиль, автомашина, автотранспортное средство. [typically can be легковые and грузовые].
slang: жоповозка (a regular car), карета (a good car), ведро (a bad car), гроб (a very bed car), дизель (a diesel drive based car), джип (a 4WD enabled car), корыто (an old car), тазик (an AutoVAZ labeled car), лошадка (a working car), старушка (a beloved car), целка (a new car, or never used/crashed car), металлолом (a crashed car), дыра (a non economic car), помойка (a cheap, very second-hand car), кредитомобиль (a Ford Focus car), пиздовоз(ка) (a car with mostly girls inside), телега (a farm car), и т. д.
also distingushed by model, limited examples only: альмерочка, альмероид (Nissan Almera), [автомобиль] какашкой (Nissan Quashqai), калека (Reno Kaleos), мерин (a Mercedes Benze labeled car), чпокус (Ford Focus), газенваген (a Voltzwagen car), и т.д.
Машина is the most common word. The word
Автомобиль is the official way of saying it. So if you would like to use it in informal conversation: use the word
Тачкаis used in jargon informal language that you might use with your close friends. Or not use at all. I wouldn't find the person who uses that word very attractive. So be careful
Колеса sounds like wheels and you should it use as often as you use this word in English. I can't say for sure but I pretty sure it's quite rare use of this word. It's fine to use these unless it has car contexts.
Колеса might also mean drugs to russian teenager's jargon unless you use the drug contexts, they might use this word in 15-20% so not that usual thing though.