You need a consistent flow of air to make paint flow consistently. Without that you will not have the control you need for various techniques. Without a tank the compressor will pulse air, so no consistency. Also without a tank the compressor will need to run continuously meaning it will soon overheat, and for safety stop working. The motor will burn out quicker.
You need a compressor, with a tank, and one that is big enough to store enough air so that the compressor is able to keep up with the demand from the airbrush it needs enough cubic feet per minute or cfm.
Depending on how you are painting and what pressure you need will determine your cfm I would say an average pressure range is between 10psi - 40psi (though could go as low as 5 or for t-shirt artists as high as 60) If you take 40psi as your probable highest pressure then you need somewhere around 23 litre a minute which I believe equals something like 1.5 cfm.