What type of foam are you talking about? Upholstery foam, also known as open cell foam, cushion foam, mattress foam, etc. it is squishy, soft, flexible foam, unlike Styrofoam which is hard and brittle. 

Where do I get foam? Well, you can get a variety of foam at department stores (such as Walmart, they sell it in their craft department as well as check other departments), camping stores, fabric stores, or specialty stores. You can use any type of upholstery foam, the only difference is the price which goes along with the density of the foam. The higher the density, the higher the price. Now, when making a costume the density does not always matter, its all personal preference. When starting out, I recommend getting whatever you can find that's within your price range. Be creative in your source, you can get flat foam mattresses (camp mattresses), from foam pillows or cushions, and I even called around and got some free from a furniture re-upholstery shop and mailing and packaging places.

How do I piece it together? Foam is pretty easy to stick together with hot glue. Just apply, let fully cool, and it should stay. If you are piecing together several layers or large surfaces (more than hot glue would be capable of applying without cooling too fast to stick) you may consider a spray adhesive (sold in hardware stores) called 3M Super 77. 

How do I trim and cut it? This part sometimes can get messy, but it can also be fun. Start with a sharp pair of scissors and a razor blade (I use a retractable boxcutter so I can have several inches of razor blade to slice with). The razor blade slices through the foam easy to "hack" out general shapes. The scissors are for rounding the shapes and trimming off edges and finishing off the shape of your forms. The messy part comes in from all the tiny snips of foam that have the potential to get all over. (It helps to have a working vacuum cleaner)

          This fleshed out enough of a shape that covered my head, and that I could wear. I marked eyeholes with my Sharpie. After cutting eyeholes out with the razor blade (where the pupils of my character would be) I could estimate where my eye shape, muzzle, and cheek placements would go by drawing onto the foam with my Sharpie. At this point, it also helps to have a drawing of the animal or intended character that you're making the mask of so you have a reference to where these facial features should be.

          I took another separate piece of the 1-inch foam and folded it into the shape of a muzzle. I just started with the crease below the nose. I did this by pulling the fold in at the angle I wanted my muzzle to be, hot gluing along the crease, and letting it fully cool so it would stay in the shape. As I got closer to my achieved shape I trimmed the foam to form the lips and folded the sides of the lips to round them off a little more, also hot gluing along the creases, holding it, and letting the glue fully cool before letting go. Once the creases where the glue has cooled and the shape is how you want it, you can trim the excess foam from the inside of your mask with scissors or a razor blade. I attached my muzzle to the rest of my form with hot glue also trimming the edges to meet at the proper angles and gluing along the edges. Folding the foam and gluing the creases also helps round out the back of the head if it comes off as too angled where you pieced the two sides together.

          The jaw in this type of mask was easiest for me to do statically. No moving jaw in this mask. It was very simple to do in one piece. Just a two inch wide strip of foam connected at two points (left side of the jaw and right side of the jaw) to the rest of the head with a small triangular wedge glued underneath for the chin. (see below for a description of a 'hybrid' technique of combining foam and plastic canvas)

          The cheeks, eyebrow ridges, and ears were all done last and cut out and shaped (with scissor-snips) as separate pieces each from thicker (3-inch thick) pieces of foam. I used my reference drawing as a guide for the shapes of all these pieces, it also helps to have a reference image from a few different angles to imagine the shape in 3-D.

          After all my pieces are all attached with glue I'm ready to fur the mask and finish it with eyes, teeth and a nose!

          I had to remove the lower jaw I had originally built and, using hot glue, attach plastic canvas to several specific spots on the foam jaw. Two lengths on either side of the of the (separate, for now) jaw that, when fit inside, would reach back to about my ears inside the mask, these were for the attachment 'pivot' points of the jaw. A length of plastic canvas underneath the jaw was placed there to help support the piece of rigid plastic I inserted in the foam (the material I used previously with the plastic canvas technique was balsa wood, but since I had leftover plastic from working with the plastic bowl technique for eyemaking, I used some from the bottom of the bowl to make the jaw more rigid where my chin pushes down.). 

          Once the separate jaw piece was concocted, I worked towards the perfect placement. Using brads (paper fasteners) I attached the two lengths of plastic canvas to the head (temporarily poked through the foam). Originally I had made the two side lengths of plastic too short, and the pivot point too much in the center. When the jaw was pushed down, it closed it more, and behaved like a teeter totter, where when I pushed it down on one end (where my chin pushed it) the other end (the supposedly open mouth) would go up more inside the muzzle instead of open! I realized my problem was that my pivot points were not far enough back. I made them longer (about to where my ears are) and then they worked good. I'd push down, and the mouth would open. To get the mouth to close again, I added a piece of plastic canvas to about the middle of where the jaw was just behind the mask's jowls and in front of the cheeks. I placed a brad there and put a tiny hair rubber band over the brad's head. I added a second brad to the matching spot on the jaw and put the other end of the rubber band over that brad's head, too. (the picture to the right shows the jaw partially opened to show the rubber band placement)

          Once I figured out the best location for the pivot points to open the jaw, and the rubber band placement to close it, I was ready to attach it to the head. I removed my temporary attachments (where brads were just poked through the foam), it may help to mark its location, and I took a rectangle of plastic canvas and poked a brad through the jaw part and the rectangle of plastic canvas, and folded down the brad's ends. I carefully glued the rectangle of plastic canvas to the head (put it back where you remember it worked the best, or where you marked it), and let it fully cool. Try and smooth out any lumps or bubbles, since this is where your jaw will be moving the most, and bubbles and lumps of glue may cause excess friction. Gluing it is a little bit tricky, since now its all attached with the jaw piece, and you want to make sure its attached to the foam well. Do this for both sides at the optimal pivot points, and then reattach your rubber bands. Try it out in the mirror, you may have to pad the inside of your mask more to get the best fit.

          This technique is also not perfect, although it will result in a working and moving jaw, you will probably have to emphasize your jaw movements as you talk to get it to show up well with your mask (this is pretty much the case with most moving jaw masks, though). Also, since foam is very flexible, I had problems with the whole head flexing when the jaw was moving, so you may need to add more structural pieces to your mask to get it so it doesn't flex as much.  Furring may fix it so it doesn't flex as much, but for this particular mask I tried this hybridized technique on, I wasn't the one furring it. Something else to note is that you want to leave a gap all the way around, you don't want the jaw to rub where it closes near the lips/jowls, remember to accommodate any fur that you may be putting there. The jaw will move more freely that way.

Front view of mask being worn, mouth closed, and side view, mouth open.