The target should be strong enough to stop arrows reliably. It's better to work with a good margin here, especially if the target is used with different arrows (carbon tend to have a lot more piercing power) or stronger bows.
Arrows should always be stopped well before the fletching reaches the target, otherwise you'll run into the danger of the fletching being damaged (oftentimes it is just shaved off cleanly).
Don't damage arrows
Obviously the arrows shouldn't be damaged, even after repeated use. This also holds for whatever construct you use to hang/install/put-up your target. E.g. using a heavy metal frame to hold up your target will mean that any arrow which accidentally hits the frame will be toast.
Ease of arrow removal
The target should be soft/flexible enough to allow easy arrow removal (if you have to put too much force into it there is always the danger of damaging the arrow).
The target should hold up for a good long while without falling apart. Styrofoam is an excellent example which fulfils most of the other requirements, but fails spectacularly on the durability (it'll dissolve into nasty tiny bits after very short use).
This concerns more how you put up your target and less how you build the actual target. The entire construct should be quite stable, so as not to be moved/thrown over while shooting at it, removing arrows or because of wind (the latter can destroy a whole bunch of expensive arrows at once).
This one is kind of obvious, but came up frequently back when I was trying to create archery targets. The proper archery foam target blocks can be amazingly expensive, not to speak of actual 3D foam targets which cost even more. Straw can be a decent and cheap alternative here, but has its own drawbacks.
Optimally a good target would resist wind and weather, which is the case for many synthetic foams.
Straw bales or designated straw targets, while cheap, don't like getting wet/moist. Meaning you'll have to manage weather exposure (e.g. cover it with a tarp, make sure it stays dry, ...).