It refers to the number of components built into the speaker. Each "way" refers to a different speaker size. Since a lone speaker cone cannot produce a flat response across all frequencies, it is more beneficial to have different size speakers. Two-way speakers will have a woofer cone and a tweeter built onto the same speaker. Three-way speakers will have a woofer, a midrange, and a tweeter. Four-ways = four speakers all on one.
The more speakers you have of a varying size (assuming you have good crossovers), you can produce a flat response across the whole bandwidth of human hearing (20Hz-20,000Hz).
A flat response means the volume level of each frequency across the band is equal to every other frequency. This is impossible with a single speaker, hence you need as many as you feel work for you.
Lesson in waves: imaging you have a 10-foot string with one end attached to a wall. You're gonna stand there and shake the string up and down to make a standing wave in the string (standing wave = it looks like the wave "bumps" are not moving towards or away from the wall, just bobbing up and down) ... you have to move the string really slow to make it work. Sure you can move the string faster and get more of those "bumps" (meaning a higher frequency), but the "bumps" are not going to be very tall.
Same effect with a big woofer...it's bigger, so it's not going to produce loud high-frequency waves. It's just not designed to do so.
Likewise with a shorter string, you have to move it really fast to make any standing waves in it. If you move it slow, you get no good waves, and if you move it too fast, the "bumps" won't be very tall. Same effect with a small tweeter...it's smaller so it has to move fast to get a clean sound. Low frequencies are inaudible from small tweeters.
Back to your question: 4-way speakers are going to be more clear throughout the bandwidth than 2-way speakers. 4-ways will be able to produce a better (flatter) frequency response than 2-ways. Notice 4-ways are also more expensive, but you get what you pay for.