Every science (and, alas, pseudoscience) is mathematized. All have their computational problems. One may wish to simulate Saturn's rings, a satellite trajectory, an electronic circuit, the vibration of an ice-laden power transmission line, the fuel consumption of a proposed design of automobile engine, an economic subsystem, the weather, the galaxy. (Or maybe just the Zodiac.) Constructing a mathematical model of your system is one thing (see the modelling section but once your mathematical equations are constructed, you have to solve them. That is, in order to produce tables of output, or more useful graphs and pictures, you have to be able to convert your problem into a sequence of good old arithmetic operations. This is scientific computing.
Once you have a scheme for doing so, you have to convert your scheme into something a computer can work with; this process is programming. Writing computer games is also programming, as is writing the web browser you're using to read this, but computers and programming were invented to solve scientific problems, and the Web and all its glory are incidental to the original purpose.
The department's (and the University's) computers are a rag-bag mix of 386, 486, Pentium, RISC workstations, Vaxen, unix boxes, and now a Cray supercomputer to replace our old Cyber. All of them are in constant use, for a real variety of problems.
All this generality might be impressive, but what about an example?
To take one at random, consider the problem of oscillation of a long, straight prism of square cross-section mounted on springs; this can be considered as an idealization of a tall, slender building oscillating in the wind, for example, from which we can hope to learn something about how real buildings behave in real winds. It turns out that after gross simplification we can write down a pair of ordinary differential equations for the motion of the cylinder and the effect on the fluid (air) surrounding the prism. That is, we can write down an equation for the position of the cylinder which depends on its velocity, its accelleration, and the velocity and accelleration of the fluid.
The equations are oversimplified, but have some hope of getting us some insight, if we can solve them. But, as it turns out, we cannot simply write down the solution---or, rather, we can write down an approximation for the solution in certain circumstances, but in fact a computer had to be used to get that approximation using a computer algebra software package; alternatively, we can use a strictly numerical technique for getting an approximate solution. Both of these are scientific computing, using a computer to solve an equation that we cannot solve ourselves without an excessive amount of tedium.
A Vancouver site with a computer algebra demonstration using Maple
The solutions have to be experimentally verified, of course, and that is what happens (among other things) at places like the Boundary Layer Wind Tunnel; but experiments are part of engineering, or physics, not applied mathematics, though the results of those experiments are crucial for the formation and verification of good mathematical models.