By applying 12 volts (or six volts if it's a six volt system) to the starter without going through the switch or any other wiring will help you determine if the starter is good or bad. I don't know which mower you have but there are two types of starters for lawn mowers. One with an attached solenoid and one with an internal solenoid. What you want to do is use a set of jumper cables to connect the solenoid or starter to the battery. For the satrter with the solenoid attached like in the photo below, find the two wire terminals on the solenoid (see photo below). One will be the ground (-), negative. The photo below shows the negative (-) has a braided copper wire attaching it to the body of the starter therefore grounding the solenoid to the starter. The other connection (+) positive, is the one without any wire attached to it in the photo below. Yours will have a wire attached to it. Probably red but, maybe not. So, black (-) to the terminal with the copper braid and red (+) to the other terminal with the red wire. They probably won't be marked (+ or -)...
This next one in the photo below has the internal solenoid and it has only one terminal (+).
The ground is achieved by bolting the starter to the engine, creating the ground path that is needed. In this case you will connect the black (-) jumper cable to the solid metal body/engine or part of the starter, not the + terminal because that is where the Red positive (+) jumper cable will go. Ensure that the (+) jumper is not touching the body of the starter, only the terminal.
Be careful not to touch the surrounding parts, frame, body or any metal parts with the cables. Be forewarned, attach the jumper cables Red (+) to the (+) terminal of the solenoid and the black (-) to the (-) terminal or body/engine ground before connecting to the battery. Connect the battery last. Don't connect the battery first. If there is gasoline residue or spillage on the engine or around the area of the starter, causing a spark by connecting the starter/solenoid last might create a FIRE!
When connecting the fully charged battery, I like to connect the red (+) first and then touch the black (-) to the battery (-). If the starter is good, you will hear the starter engage and turn the engine over. If it does not make any noise whatsoever, check your connections. Still no noises, you may have a bad starter. If you hear a clicking sound or if it sounds like it's trying to turn over but not quite, check to see if the connections are good and the battery is fully charged. Some times tapping on the body of the starter with a hammer handle (HANDLE) will get the starter to turn.
That's basically how I check to see if a starter is good or bad. Another way is to take it out and bring it to a mechanic shop and have them test it. Some will for a small charge.