As you open the hive, pay attention to things like odor, activity, and anything else in the hive (especially if there are other things in the hive - mice, beetles, larvae in weird places, etc. may be immediately indicative of a problem or secondary to something else going on in the hive).
Make a habit of counting the number of frames covered with bees in each box - this is easily done from above by counting the seams between frames that are full of bees.
Burr comb (filling spaces that are bigger than "bee space") and brace comb (going across spaces between frames) are worth noting as well. your hive is generally easier to maintain if you scrape burr and brace comb off when you have a chance. If this comb is left long enough, the bees will fill it with food or brood.
As you go through frames, check each side for pollen, capped honey (white "dry" caps or dark grey "wet" caps), uncapped nectar, and the various stages of brood. Of course, it is always a great feeling to see the queen, but sometimes this can be difficult.
Sometimes the queen is less visible, but the worker activity and movement around her gives her away - the circle of attendants is known as her retinue.
When you can't find the queen, evidence that she is there and doing her job is often sufficient. Look for small larva and brood chamber cells with one egg per cell placed neatly in the center of the cell. These eggs will gradually lean over before hatching. Larger larva are good as well, but the very young larva (less than 3 days old) and eggs are a resource - if the hive has to, they can choose one of those to raise a new queen.
You may see eggs that are less perfectly placed - this is usually not ideal. The picture below is from a young queen that was just learning how to do her job and laid two eggs in a few cells. If you see many misplaced eggs, you should be checking to see if you have a laying worker in a queenless hive. A laying worker can only lay drone eggs and the hive will die without a continued population of worker brood.
The hatched larva will be fed and cared for by the nurse bees (young adult worker bees) and when they are ready to pupate, their cells will be capped. Worker brood will have flat tan/brown caps. In this picture, you can see the progressive brood pattern (capped brood surrounded by progressively younger larva and a pollen border on the edges of the brood chamber).
Drone brood will have muffin-top-shaped tan/brown caps and will often be found in "patches" of comb with larger diameter. Such a patch of drone brood can be seen below with uncapped nectar in the nearby cells.
As the season progresses, you should see a radiating pattern of brood laying and hatching develop within your hive - ideally there will be a ring of pollen around the outside of the brood chamber and capped honey in the top corners of the frames as well. The frame in the picture below is from the center of a well-developed brood chamber mid-summer.
Pictures Submitted by KT Thompson