I'd like to run two 20A circuits to the other side of my front yard. I'll be using 12AWG THWN conductors, over a length of about 20 feet through PVC conduit buried at a depth of 18 inches.
What diameter conduit is appropriate for my application?
Code defines minimum sizes, based on fill percentage. Larger also meets code (by exceeding it.)
Anyone who has ever pulled a conduit at/near maximum fill % (40 for 3 or more wires in conduit longer than 24") can tell you it's not fun to do.
You have a short run across the yard, so the conduit expense is not a huge factor. In general, the conduit expense is always much, much less than the trench to put it in expense.
I would use schedule 80 PVC for a job like this; while it will mean finding your local electrical supply unless you have the oddest home cheapo I've never met (never seen sch. 80 at any of the ones near me, other than the odd sweep) it is far more robust than schedule 40. I would also use 1 inch - not because "code says you need it" (it does not) but because it is somewhat stronger and provides considerably more room than you require - and more room makes pulling wires easier. It also means you don't really have to define exactly what you are doing (so the absolute minimum size you need can be specified) because it's more than large enough for considerably more than you have stated no matter how you do what you've stated.
I would (but you don't need to) put in a second 1" conduit while the trench was there, if there was the remotest chance on the face of the planet that I might ever want to run phone, network (perhaps an outside wireless access point), garage door opener or any other sort of low voltage wiring to the location. See cost of conduit .vs. cost of trench again. But it's your call.
Finally, don't skip the "buried electric line below" warning tape in the top 6 inches of fill for the trench. Digsafe won't know your private line is there, so they won't mark it.
You could do two 20 ampere circuits, or one 20 and one 15 ampere circuit. If you do two 20, you could run a multi-wire branch circuit, which would save you a wire.
Since you're using conduit underground, you'll have to use wire rated for wet locations. You'll probably want 12 AWG THWN wires for 20 amperes, or 14 AWG THWN wires for 15 amperes.
You'll need white or gray wire for the grounded (neutral) conductor(s), green for grounding, and black, red, blue, etc. for ungrounded (hot) conductors.
If you're sure you'll never want to run any more circuits through the conduit, 1/2" schedule 80 PVC will fit six 12 AWG THWN conductors. That should be plenty for your two circuits, even if you opt not to have a MWBC. If you might run more circuits, or you want an easier pull, you could install 3/4" schedule 80 PVC.
You'll want to bury the conduit 18" deep, and provide GFCI protection for the circuits (most likely via GFCI breaker(s)). If the circuits are 20 amperes or less, and are GFCI protected, you only have to bury the conduit 12".
Keep in mind that this is a very broad overview, as you haven't provided much detail in your question.