MacBook – 1920×1080 resolution not in the options after connecting external monitor on 2015 Macbook air running El Captain

With the Display pane of System Preferences visible, hold down the Option key and the "Gather Windows" button in the lower right should turn into a "Detect Displays" button. Click on that button and your computer should take care of the rest.

You can get full resolution with mini DisplayPort to DisplayPort or mini DisplayPort to Dual-link DVI. Dual-link DVI supports up to 2560x1600 pixels resolution. DisplayPort supports that resolution or even higher.

You will not get full resolution with mini DisplayPort to DVI. DVI is limited to 1920x1200 or something similar. You usually won't get full resolution with mini DisplayPort to VGA.

I do not know whether you get full resolution with mini DisplayPort to HDMI. I think it depends upon your graphics card.

Bottom line. I would recommend using mini DisplayPort to DisplayPort.