Polygon count
The first constrain is polygon count, This is more of a constraint for 3D real time programs as oppose to say a 3D animated movie. This is because the movie can be rendered out and will then have all the polygons pre rendered where as in 3D real time programs such as video games the user is free to look where ever they wish meaning that the hardware has to render the polygons dependent on where the character is looking.
Therefore this is a huge constraint as the actual performance of your video game may suffer.
I managed to find information on the poly count that was used for crysis. This game was released in 2007 and was a ground breaking game for its time with some of the highest quality textures, lighting and character models this game was often used just to see how powerful your PC really was.
Characters
Heads 3,000 triangles
Body 5,000 triangles
Weapons
First person: about 6,000 triangles Third person: about 1,500 triangles
However if you have ever played video games you will have noticed that the cut scenes are always so much better looking, well this is due to the fact poly count is less of an issue (still a constraint) as the polygons have already been pre rendered however a larger constraint for cut scenes as well as the actual game play is file size.
File size
File size is very important especially for video games as they are often put on to disks with limited storage capacity on them meaning that they have to get everything that the game will include and compress it onto a single disk however as games are becoming more advanced in technology they have started having to put it on two disks due to the amount of information in the game. This is also why PC games look better graphically than console games as PC you can just download all the information as well as not being restricted to hardware restraints unlike console. however there are still restraints even on PC as to how large things can be due to the hardware having to read all the info and so larger file sizes often leads to more work on your CPU.
Things to reduce file size
Change render resolution
Reduce the size/reselection of textures
Use a modifier called "ProOptimizer" (this reduces the vertices)
Do not keep modifiers on objects once finished modifying
Rendering time
Render time is especially important to company for example Pixar can spend years rendering their latest movies and this is even done through render farms (basically a load of computers solely for rendering. There are a few ways in which you can decrease this time however the options you change to speed up render time will ultimately affect the look of the render.
If your render doesn't require reflections then you are probably going to want to turn ray tracing off. This basically creates mathematical formulas to replicate realistic lighting which is very powerful and power consuming on your CPU.
Another way to cut down your render time is to turn down or off anti-aliasing this basically smoothens the model so you don't get jagged edges in the rendered image.
