Some warning signs I've seen when house shopping -
Shoddy construction. I've seen a lot of houses with very poorly constructed roofs over porches, finished basements, etc, which clearly weren't inspected. Some cities will go after the current home owner for non-permitted improvements, even if the work was done 4 owners ago. Dealing with the work - either fixing it up to meet basic standards, ripping it out, or cleaning up after it falls apart - can be a headache as well.
Cut corners. One house I saw was a new remodel but there were pieces of floor missing in some of the corners, the kitchen cabinets didn't quite fit, the counters were poorly mounted, etc, so I had to wonder - what issues couldn't I see?
Water damage. This can be tricky because sometimes it's hard to tell if an old appliance leaked or if the basement is prone to flooding in a storm, but any sign of water where there shouldn't be and I'll want to investigate.
Foundation problems. One house I saw had cracks on both ends going from the bottom to almost the roof - in both the drywall and the brick exterior. Another had pieces of the foundation falling off into the crawlspace. In general, a vertical basement crack that has no "movement" - that is, both sides of the crack are even with each other - doesn't indicate a problem. A horizontal crack, or a crack where one side is further out than the other, is more likely a problem. Rooms with slanting floors can also indicate a problem.
Roof problems. If it's an asphalt shingle roof, how many layers does it have? If you look at the roof on edge, can you see any bowing or sagging? Is the ridgeline still straight? If there's a chimney, is mortar missing from the brick, is any flashing pulling up, etc. Look under the gutters, can you see an places where water has been dripping? Do the soffits look secure or are there holes where wildlife could get into the attic?
Odor. Some smells are hard to eliminate - cigarette smoke, cat urine, etc. These smells tell me that I'm going to be removing carpet, painting walls and more if I buy the house.
Then of course there's the obvious - how much work does the landscaping need, how old are the appliances, is the carpet in decent shape, how beat up are the walls and doors, is the kitchen a hideous tribute to the worst the 50s had to offer, does the bathroom make you run in horror, so on and so forth.
Any of the above issues will affect the value of the home. When I make an offer I like to take the price I'd pay for that house in good condition, subtract the cost of materials needed for improvements, subtract some extra for my time (or labor if I can't do it myself), then a little more extra for overages/unexpected issues that might arise.
And, of course, you should almost always make a bid contingent on inspection then hire a pro to check out the house.