A couple of weeks ago when I made the black and white cake, I took a bunch of pictures while I was working on the bottom cake tier to share here how I usually go about filling and icing a layer cake.
1. Start by baking a cake. I prefer to make round layer cakes just because they are easier to ice since they don't have corners. I bake all of my cakes at 325 degrees because they rise more evenly and don't typically develop that "dome" or "hump" on the tops, that I would end up having to cut off and usually throw away. For layer cakes, I bake three cakes, but I've also been known to just bake two oversized cakes and cut those in half to end up with a four layer cake instead of three layers. After I bake the cakes, I let them cool on the counter for around 30 minutes to an hour, and then I typically will let them cool completely in the refrigerator overnight.
2. Make your icing, and make it pretty thin so that it is easily spreadable. Take one of your cakes and turn it out of the pan so that the bottom of the cake is now the top and put it on a cake plate or cake server. Since I knew I was going to be stacking 2 more cakes on top of this one, I placed mine on a large, sturdy cake plate made from heavy duty cardboard and styrofoam, also known as a cake drum. If your cake has a dome or hump shape to it, you will need to trim the top off so that the cake layer lays completely flat on the plate. Obviously you will have to do this before you flip the cake over so that the bottom side is facing up on your plate. Usually I will put a little icing on the cake plate itself before putting the cake on it, so that the cake has something to adhere to. Once you have the first even cake layer down on the plate, put a good dollop of icing down in the middle of the cake layer. For a 10 inch cake, I'd say I probably put about a cup of icing in between each layer. Moving from the middle outward, spread the icing out to the edge of the cake layer. In order not to get crumbs into the icing, your icing should be pretty thin because thicker icings will pick up and drag more crumbs as you are trying to spread it out.
3. Place your next cake layer on top (bottom side up) of the iced layer and repeat the process with the icing filling. Finish by placing your last cake layer on top (again, bottom side up). Put the cake in the refrigerator for at least 30 minutes, I usually let mine stay in the fridge for at least an hour so that the icing has a chance to harden up some. This step makes it SO much easier to proceed with the following steps, trust me! Otherwise, if the icing filling is still pretty soft and you start putting your outside icing on the cakes, the cake layers will tend to slide around making it frustrating and nearly impossible to ice the cake properly.
Rather than try to describe every detail of what I did next to ice the outside of the cake, I figured it might be easier to actually SHOW you a video of someone doing the technique it would take me forever to describe in words. The only difference in how I ice a cake and how this woman does it is that I don't do the "Viva papertowel technique". Rather, I use a small paint roller with a sponge roller on it (pictured) and it works actually much better than the papertowel technique like she is using in the video. After I ice my final layer of icing on the cake, I let it sit on the counter for about 20 minutes until the icing starts to "crust". If you tried to smooth it right away, the icing would be too sticky and would stick to the roller and generally be a giant pain in the neck.
I also start spreading my icing at the top of the cake and work my way on the sides, but everything else is basically the same. One day I hope to actually get some software for my computer to make my own videos, but for now, I hope this information is helpful! The woman in the video refers to a "crumbcoat" a couple of times, and basically all this is is a thin layer of icing put on the cake and then let harden so that when you put the second layer of icing on the cake, you don't have any visable crumbs so that your cake is smooth and pretty!

Please let me know if you have any questions on this tutorial, and I hope you enjoyed it!