MOST importantly, please do not buy a notch lapel dinner jacket (tuxedo). Dinner jackets should be shawl or peak lapel only, a notch collar is simply an abomination.
to answer the rest of your question: A tuxedo consists of a jacket (usually in black, sometimes white, midnight blue or a color if a more informal version) which has (generally) satin or grosgrain lapels, cloth-covered button(s) and a matching pair of trousers with a detail down the outseam matching the lapels. The most common classic dinner jacket ("tuxedo") is a one-button single breasted version, and there are also two to three button single breasted and six button double breasted, shawl and peak lapel versions.
Wearing the whole outfit without the cummerbund or waistcoat would not work. Wearing the pants or the jacket with other parts, if you knew what you were doing, could work. Personally I have been looking for a good pair of tux pants to wear with other stuff. A dinner jacket with beat jeans, etc. has long been a fashionable possibility, although you really do have to know what you're doing to pull it off. Wearing the whole thing without actually having a reason simply would not do.