What type of weave is this?
The front looks like standard knitting (stockinette stitch).The picture quality of the back is too low to say for sure, but it looks similar to how the backside a stockinette stitch should look.

The stockinette stitch is the most basic stitch and used for most machine-knitted fabrics. You find it in almost all t-shirts, sweaters, socks and other knitwear. 

The notable difference of your picture to that of most knitwear is that the right side of every stitch is twined so much that it appears to be one continuous line. This is a sign of quality because the single fibers of a yarn cannot be pulled from it easily if it's twined.

So my guess is this:
The fabric is knit by machines in the stockinette stitch like almost all knitted fabrics, but the manufacturer used high-quality twine instead of cheap thread.

For comparison: This is how a stockinette stitch looks like with yarn that was not twined. The regular V-shapes of the individual stitches are clearly visible.

Image source  

And this yarn was slightly twined. It's the same pattern, but the stitches seem to blend together into rows of vertical and diagonal lines.

Image source