You're envious of your neighbors lawn. He has a perfectly level cut like this. And you want to know how to get it.
First, it starts with the quality of your turf. You have to invest in a fertilization program for your lawn. It will take several months of care and investment to get your lawn to full, thick, and healthy status.
Also if you're in cool season grass zone and have tall Fescue for instance, you'll need to aerate and overseed your turf every fall. Without a thick healthy lawn you wont have anything to work with when it comes to mowing it beautifully.
Next, make sure your blades on your mower are sharp. You will need to sharpen them every 15-20 mowings or at least twice per season. I recommend sharpening them first thing in spring, and then again in mid-summer. During the summer months is when your lawn is most susceptible to damage from heat stress, so its imperative to have sharp blades cutting the grass.
After you have a good sharp blade, scrape out the deck of your mower removing the build up that accumulates underneath. You can use a regular painters spatula for this task. Typically, you will need to clean out the deck of your mower every other cut. This allows your mower to do its job better, and yields a better cleaner cut. Put a plastic trash bag down and scrape it out over the bag, this makes for easy cleanup of the debris.
Now check over your lawn very closely looking for divets, pot holes, and imperfections in the grade. This is crucial because if you have uneven terrain your mower will scalp areas as you are cutting the lawn. This is particularity the case if you have a riding lawn mower as they are not very forgiving and will case damage to the lawn as they run through a pot hole.
Following this advice you too can create a beautifully shorn lawn that looks like glass... like this..