The Thunderbolt 2 port on that MacBook Air uses a Mini DisplayPort connector, and doubles as a Mini DisplayPort. (The Mac senses what type of device is plugged in, and adjusts the use of the port accordingly.)
Most modern monitors have (Mini) DisplayPort, HDMI, and/or USB-C (DisplayPort) inputs. I have not heard of any adapters to go from Mini DisplayPort output (on a computer) to USB-C (DisplayPort) input (on a monitor). (There are more USB-C to Mini DisplayPort adapters than you can shake a stick at, but they go in the other direction.)
So you want a monitor with DisplayPort, Mini DisplayPort, or HDMI input. (It's OK if a monitor also has USB-C input … just as long as it has at least one input that your current Mac can use.)
DVI and VGA are legacy inputs – so if a monitor only offered them, I'd suggest skipping it, and moving on to something else.