The phrase geographic coordinate transformations refers to the conversion of data from one datum to another. Many GIS software packages have the ability to convert data from one datum to another. However, you may find yourself in a situation where the software conversions that are readily available will not solve your datum conversion problem. For example, the transformation you want to perform may not be a standard option, or you may not know what datum your data are in (this is of course necessary for choosing the correct transformation). In a pinch like this, it is still possible to convert your dataset from one datum to another by using what is known as an affine transformation. You may have first come in contact with the affine transformation in the context of georeferencing scanned or tablet digitized data. Strictly speaking the affine transformation is NOT a datum transformation.
The affine transformation includes three types of operations: translation, rotation and scaling of one datum to produce a match to a second datum (see Figure 7.cg.8, below). In order to successfully convert your dataset, you will also need to know the locations of at least three points (preferably more) in both coordinate systems. The more points you have, and the more equally distributed the points are across your area of interest, the better the transformation will be (i.e., you will be less likely to introduce positional error into your dataset by performing the transformation).
The term "projecting-on-the-fly" refers to a GIS software feature that was designed to take unprojected datasets (i.e., data that are stored in latitude and longitude) and transform their visual representation to make them appear to match other datasets that are projected. The capability has evolved in some GIS programs to one that can re-project from one already-projected coordinate system into another. (Though behind the scenes there is an intermediate step whereby the coordinates of the initial dataset are un-projected to lon-lat, then projected to the desired second map projection.) In this method of displaying data, the underlying coordinate system and projection of the dataset are not altered. Although taking advantage of this feature may save you a few steps and make it easy to make preliminary visual inspection of a dataset, it can cause problems if it is used incorrectly with datasets that are stored in two different geographic coordinate systems; it is easy to apply an incorrect datum-level transformation to a dataset. For this reason, the best practice is to always transform all of your datasets so that they are stored in the same coordinate system and map projection.
Reprojecting your data, in contrast to projecting it on-the-fly, does make changes to the actual data (i.e., both the dataset and the visual representation of the dataset are altered). When reprojecting your data, you may find that you have to first unproject your data (i.e., transform them into latitude and longitude coordinates) and then reproject them into another map projection. This will depend on the capabilities of the GIS software you are using.