Advance Width is a metrics term related to fonts/typography. There are some Q&A's here on SO that cover it, but the short definition is more or less:
The advance width is the distance between the glyph's initial pen position and the next glyph's initial pen position.
A visualization usually makes things easier to understand, so have a look at the image below.
Image courtesy of freetype.org.
Similar to the advance width, there is also a less frequently used advance height:
Image courtesy of freetype.org.
Freetype's article on glyps metrics goes more in depth about the various metrics. Might be worth a read if you're interested.
Note: I'm in no way affiliated with Freetype.org. I just happen to find their documentation on this matter useful.