(From: David Thomas (dthomas@NO.cityutil.com).)
The following assumes it is (was) a small push mower with a vertical shaft engine and that you drained the oil the preceding fall.
What little oil that was left in the bushing areas on the crank have burned down to tar. The piston may or may not be seized. Either way you'll need to break the engine down (pulling the flywheel is the hardest thing about it) and clean the tar out.
If this is beyond your abilities, try spraying penetrating oil on the crank around the bushing areas both on top (may still have to remove the flywheel to get to this area) and bottom as well as around the piston as the other gentleman suggested. (If you can't remove the head, pull the spark plug and spray penetrating oil or WD40 inside the chamber using one of those straw things that comes with the can so that you can direct the spray around all the walls of the cylinder, put the plug back in so it doesn't all evaporate and then set the engine so the piston is pointed straight up and let it soak a few hours.)
Last choice is to fill the crankcase with about a pint or two of penetrating oil (and spray inside the combustion chamber as stated above) then slosh it around the engine, turning the engine every which way and upside down, let it soak for a couple of hours repeating the sloshing every so often and changing the position of the engine so that you alternate soaking each bushing area and the bottom of the piston and then try the pull rope again. Be SURE to disable the spark 'cause you SURE don't want it to start with that light oil in the crankcase.
If it does free up, drain out the light oil and add the normal recommended oil before trying to start it. If it starts (it will smoke like the devil while it burns that light oil), let it run at idle speed until it warms up then shut it down and change the oil. It may still burn oil and smoke since the piston walls are probably scored badly so check the oil every time before starting it even after a short break until you get a feel for how much its going to burn.
(I had one engine that I did this with that used all most as much oil as it did gas. I had to clean the crud out of the spark plug before each mowing.)
(From: Brian Fistler (firstname.lastname@example.org).)
If it's a Tecumseh engine, throw it away... Even if you get it running, it'll probably throw a rod soon (not like it wouldn't even if you hadn't seized it... :-)
If it's a Briggs, try putting oil in it first, then remove the spark plug and spray a liberal amount of WD 40 in there... You might have to let it set for a day or so, then, LEAVING THE SPARK PLUG OUT try turning the engine using the blade... Once you get it freed up, most likely it's run fine and probably last a few more years...
When I was a kid, my best friend and myself used to "work" on Briggs engines, we'd tear them down, fix broken ones from the junk yard, and most fun of all was "torturing" one... After seizing it up by finding out what would, and would not keep the thing running by spraying it in the carburetor (i.e., using hair spray, WD-40, paint thinner, etc...) We finally found that a certain spray engine degreaser of the era WOULD run the engine, it would not run it for LONG because it removed the oil from the cylinder walls, and the engine would seize up...
That gave us an excuse to tear the engine down and find out how to get it going again... After the 3rd or 4th time seizing the engine with various products, we just decided to cut a hole in the side of the block and put a piece of Plexiglass there, so we could get easier access...
We ran that poor engine with every possible thing in the crank case, including pure water... (Side note: Water didn't seize the engine, as long as you didn't allow all of it to evaporate)
When the engine would seize up, all we ever had to do was put oil back in the case, spray a little WD-40 in the spark plug, and hit the blade a few times with a hammer to get it starting to turn...
That motor kept us entertained for an entire summer... :-)