A coaxial speaker is just any speaker that combines the tweeter(s) and bass into one speaker. You've heard of component speaker systems right? Component systems are systems you can buy where every aspect of the system is seperate (i.e. the tweeters, mids, and bass are all seperate speakers and usually the tweeters come with there own crossovers).
Coaxial speakers are "all-in-one" and are typically cheaper than component systems. You can have either "n"-way coaxial speakers or "n"-way component speaker systems but it all depends on which you prefer.
Like Haulin_A_Doo said a "n"-way speaker means there are "n" speakers together. For instance a 3-way will have a tweeter for highs, a mid for middle notes, and a sub cone for bass. A 4-way typically has 2 tweeters instead of one. Whether you want them all seperate or "all-in-one" will determine whether you buy a component system or coaxial speakers.
Hope that helps