Eigenface might be a good algorithm to start with if you're looking to build a system for educational purposes, since it's relatively simple and serves as the starting point for a lot of other algorithms in the field. Basically what you do is take a bunch of face images (training data), switch them to grayscale if they're RGB, resize them so that every image has the same dimensions, make the images into vectors by stacking the columns of the images (which are now 2D matrices) on top of each other, compute the mean of every pixel value in all the images, and subtract that value from every entry in the matrix so that the component vectors won't be affine. Once that's done, you compute the covariance matrix of the result, solve for its eigenvalues and eigenvectors, and find the principal components. These components will serve as the basis for a vector space, and together describe the most significant ways in which face images differ from one another.
Once you've done that, you can compute a similarity score for a new face image by converting it into a face vector, projecting into the new vector space, and computing the linear distance between it and other projected face vectors.
If you decide to go this route, be careful to choose face images that were taken under an appropriate range of lighting conditions and pose angles. Those two factors play a huge role in how well your system will perform when presented with new faces. If the training gallery doesn't account for the properties of a probe image, you're going to get nonsense results. (I once trained an eigenface system on random pictures pulled down from the internet, and it gave me Bill Clinton as the strongest match for a picture of Elizabeth II, even though there was another picture of the Queen in the gallery. They both had white hair, were facing in the same direction, and were photographed under similar lighting conditions, and that was good enough for the computer.)
If you want to pull faces from multiple people in the same image, you're going to need a full system to detect faces, pull them into separate files, and preprocess them so that they're comparable with other faces drawn from other pictures. Those are all huge subjects in their own right. I've seen some good work done by people using skin color and texture-based methods to cut out image components that aren't faces, but these are also highly subject to variations in training data. Color casting is particularly hard to control, which is why grayscale conversion and/or wavelet representations of images are popular.
Machine learning is the keystone of many important processes in an FR system, so I can't stress the importance of good training data enough. There are a bunch of learning algorithms out there, but the most important one in my view is the naive Bayes classifier; the other methods converge on Bayes as the size of the training dataset increases, so you only need to get fancy if you plan to work with smaller datasets. Just remember that the quality of your training data will make or break the system as a whole, and as long as it's solid, you can pick whatever trees you like from the forest of algorithms that have been written to support the enterprise.
EDIT: A good sanity check for your training data is to compute average faces for your probe and gallery images. (This is exactly what it sounds like; after controlling for image size, take the sum of the RGB channels for every image and divide each pixel by the number of images.) The better your preprocessing, the more human the average faces will look. If the two average faces look like different people -- different gender, ethnicity, hair color, whatever -- that's a warning sign that your training data may not be appropriate for what you have in mind.