Pretty much what everyone else wrote. These were our personal considerations that we predicted and which worked out great with our current house, just as examples. Side note: we knew exactly what we wanted because we were in our 40s and had lived in a lot of different places/houses.
General location: for us, we wanted an older, residential, upcoming neighborhood with sidewalks, house near small shops, a library, grocery--schools didn't matter to us, we're DINKs. We did not even look at houses that were NOT walkable to a metro stop and/or bus stop and near enough to DC and to DH's job. Walkability/public transport was that important to us. Neighborhood near DH's work (I didn't have a job yet, but it ended up near my work as well). That is, I jumped at a job that had an office near our home.
Size and layout of house: Detached house, not a townhome. Not too large for our needs, only the number of rooms we really use everyday, layout is open in the public rooms. Nothing freakish in the way you access places in house. When looking at houses, we saw some crazy steep/narrow stairways, bedrooms only accessible through a kitchen; the layout should make sense and you can imagine yourself getting up, moving about your day.
House condition: Structure and basic systems need to be in good shape and not too old. For an a/c system that had some problems (which the sellers disclosed), we got a quote and asked the sellers to take that amount off the price (they did). We didn't want to have to redo everything, so we were happy to see a recently redone kitchen. On the other hand, we actually were HAPPY to see a really nasty bathroom because it gave us a chance to gut it and put in exactly the bathroom we wanted. Keep an open mind on condition.
Lot size and location: Not on a main road, not noisy, enough parking on street for visitors. We have a short driveway, so don't have to park on street. Lot has a nice set-back for front yard, not too close to sidewalk, backyard is large and gets enough sun all day to grow vegetables (if you are a gardener don't buy a place where you don't have any sun). Drainage: lot is on top of a slight hill, not at the bottom of the hill. We thank our stars for this all the time (there have been some issues with flooding down the street).
Not so important, what I didn't predict: minor aesthetics, paint colors, "ugly" things. Somehow over time the things I was sure I would replace have just not offended me anymore (or I can live with them). We replace things when they cost us in maintenance or problems. The roof was leaking, it had to be replaced; those ugly plastic shutters-nah, they stay.