Often we know what the dimensions (or units) of an answer to a problem should be. Checking to make sure our answer has the dimensions (or units) we expect is useful. If the dimensions are what we expect then our answer might be right, if they aren't then our answer is definitely wrong.
Example: The dimensions of volume are Length^3. This is often written V [=] . If I told you that the volume of an ellipsoid was V = where 2a, 2b, 2c were the lengths of the major axes of the ellipsoid you would know that my formula was wrong because [=] .
Dimensions usually refer to fundamental quantities like mass, length, or time. Units can refer to these dimensions or to derived quantities.
Examples: Meters and inches are both units of length. Force has dimensions of (mass)(length)/(time^2) and has units of Newtons in the SI (Système International).
Note that if I am adding or subtracting quantities they have to have the same units. Furthermore, expressions on each side of an equal sign must have the same units.
What about transcendental functions like Sin[x], Log[x], or ? The arguments of functions like these must be dimensionless! If x was length then Log[x] has no meaning. We know that the trig functions take arguments in radians. Radians are dimensionless. If the arguments of these functions must be dimensionless then the results from evaluating these functions must also be dimensionless. If one has a physical problem where x [=] L that leads to a formula involving Log[x], for example, then x must be redefined as a dimensionless variable. This is usually easily done, for example define = x/a where a is some naturally occurring length scale in the problem.
How do the basic operations of calculus change the dimensions of an equation?
Examples: Consider distance, time, and velocity: x [=] L, t [=] t , and v [=] L/t. We know that v = , thus must be [=] 1/t. Similarly the integral of velocity with respect to time = x + C, thus dt [=] t.
If the differential dz represents the limiting difference of some quantity z, then dz will have the same dimensions as z. You might want to think of this in terms of what the derivative represents geometrically. Similary, thinking of integration as the sum of rectangles of base dz and height f[z] shows us that integrating f[z] gives a result with the dimensions of f[z] multiplied by the dimensions of dz.
Example: A careful reader of a Calculus textbook would check to make sure that the formulas for the volumes of solids of revolution were dimensionally consistent. In fact, thinking dimensionally would be an aid in remembering the formulas (or even better yet, the derivation of the formulas!).
Note that the fundamental dimensions generally are length, time, mass, temperature, and electric current. One could (but we usually don't) define a different independent set (for example using electric charge instead of electric current). One can (and often does) use different systems of units such as British versus SI. Units can be defined for convenience, for example "apples" and "oranges" are OK (but not of interest in physics!). This paragraph could be summarized by saying that extraterrestrials would use the same (or equivalent) set of fundamental dimensions, but would use different systems of units.
Example: Force has units of Newtons in the SI system. Force does not constitue a new fundamental dimension since it can be derived from the set above: Force = (mass)(acceleration), thus the dimensions of force must be mL/. In different systems of units this might be given different names such as dynes, Newtons, or pound-force.
Dimensional Analysis usually refers to the process of a) putting equations in dimensionless form so as to be as simple as possible, or b) finding the fewest possible "dimensionless groups" for correlating experimental data. If one cannot solve (or even if one does not know!) the governing equation for a process, but the list of relevant variables is known (or can be guessed), then the minimum number of dimensionless groups can be found from something called the Buckingham Pi Theorem. For further reading consult a) engineering textbooks on fluid mechanics or heat transfer, or b) textbooks on analysis of experimental data. Dimensional analysis is a powerful tool for simplifying equations or experimental data that arise from complex situations.