What is a chopped or bottomless portafilter? It is one where someone quite literally cuts out the bottom of the portafilter so that the coffee comes out of the basket and falls straight down into the cup. This may seem like a crazy idea and in some circumstances a bad one, for example, stopping you from splitting the shot, but there is some method behind the madness.
The biggest benefit to it is the clarity of view of the espresso as it falls from the portafilter. It gives the aspiring barista an unobstructed view of their shot revealing both its beauty and its flaws. If you have not packed the puck evenly or well enough there will be clear indications. Spritzes, channeling, and donut extractions are all indications that something has gone wrong.
Now this method is not foolproof. It is possible with some cunning to pull an absolutely vile shot of espresso that looks stunning out of a bottomless portafilter. Raise the temperature too high, for example, and you can make an insanely bitter shot that looks good. I have also tasted shots that by looking at them from a bottomless portafilter should not have been good, but in fact were quite good. With that said, I would argue that the average shot that looks good coming out of a bottomless portafilter is going to be much better than the average shot that does not look good.
I also think (although I have not tested this) that you can taste the difference between shots pulled on a bottomless portafilter from a spouted one. The bottomless tends to produce more crema, probably because there is less friction popping the bubbles as they fall freely into the cup instead of running a longer path along the metal. I do not prefer one way over the other for taste and I quite enjoy the subtle difference. Sometimes I am in the mood for one and sometimes the other.
If I could only have one portafilter it would certainly be bottomless as I want this when I am first dialing in a coffee. You can buy them online through any number of vendors, you can send off your portafilter to have it chopped, or you can do it yourself. One more topic for later is a how to for chopping your own portafilter (it is not hard to do).