When air is heated it either increases in volume, increases in pressure or both, depending on the conditions. Air is composed of several different gases, most notably nitrogen, oxygen and carbon dioxide, but the particular gas involved does not matter. All gases behave identically with an identical increase in temperature, as long as the other conditions are also identical.
Whether air expands or increases in pressure depends on whether it is restricted in any way from expanding. Expanding air also decreases in density. A sealed container of air cannot increase in volume when heated, so it only increases in pressure. Theoretically, in an unconstrained situation air only expands when heated, but in reality, since even surrounding air masses provide some resistance, an increase in temperature always creates at least a slight increase in pressure. A more significant example of both volume and pressure increasing would be a rubber balloon. Heating air in a balloon increases both pressure and volume, since, while the balloon is elastic and does expand, it also provides some resistance to that expansion.
The fundamental equation for exactly how a gas or mixture of gases behaves when heated is PV = nRT, where P = pressure, V = volume, n = the number of moles of gas molecules, R is the ideal gas constant and T = temperature.