The first thing we remember when thinking about the physical lift force is probably aircrafts, though lift can be generated by any object there's no greater illustration than a huge airliner flying high in the sky helped by two airfoil shaped wings. Having said that, informed geeks should be aware of the physical laws governing such interesting phenomena so that they can be better appreciated or correctly explained to their friends. Obviously explaining lift involves some math but this article will just give you some directions from which you can develop into a more advanced understanding by doing your own research.
There are basically two ways of explaining lift, one uses Newton's laws of motion while the other uses the Bernoulli's principle. Though both are equivalent I'll start with the first since Newton's mechanics are definitely easier to grasp having most probably been your first contact with physics, on top of that the Bernoulli's principle requires some knowledge of integrals and differential equations. The complexity of fluid dynamics has in fact lead to several misconceptions, particularly concerning the Bernoulli's principle approach resulting in totally erroneous explanations of lift. The most famous one states that asymmetric airfoils where one surface is longer than the other produce lift due to the so called "equal transit time", the fluid flowing on the longer surface would have to increase its speed to reach the one on the shorter surface at the same time creating a pressure differential. Unfortunately many authors are still selling books using this fallacy.
Newton's second law of motion states that a force equals mass times acceleration (F = m * a). Acceleration by its turn equals the derivative of velocity with respect to time (a = dv/dt), put into other words an acceleration exists if a body experiences a change in velocity during a certain time interval, the greater the velocity change or the shorter the time interval the greater the acceleration. Any object immersed in a fluid such as air or water can be inclined producing a deflection on the flow direction of the fluid, alternatively the object may have a special shape which also creates this deflection. As a result the velocity of the fluid changes, remember that being a vector quantity the velocity may change in magnitude or simply if the movement changes direction keeping the speed magnitude constant. Thus as a result of the change in velocity we get an acceleration (a = dv/dt) and consequently a force (F = m * a) as seen before, this force suffers a reaction with the same magnitude in the opposite direction (Newton's third law of motion) which has the special name of lift.
The Bernoulli's principle states that an increase in the velocity of a fluid implies a correspondent decrease in pressure. When objects are immersed in a fluid the application of mechanical forces occurs through all the surface of the body rather than at a single contact point since the fluid covers the object all around. Pressure is the perpendicular force per unit area applied on a surface (p = Fn/A). Thus there's a pressure the fluid exerts on the object at every single point of its surface which acts perpendicular. One way to obtain the resulting lift force applied to an object immersed in a fluid is multiply the pressure by the area (Fn = p * A) at each distinct surface around the object and integrate over all the surface. Now picture an airfoil or simply a flat object which is static in the air, since the air will exert the same perpendicular force over all the sides they will all cancel out and the resulting lift force will be zero. However, once the object is in motion the velocities of the fluid around each surface will vary and consequently different pressures will act upon the object as stated by the Bernoulli's principle. This means the resulting lift force will now exist because the perpendicular forces will be different not canceling out.