Congrats! You just got a new handgun. I hope that you did your research and got something that you love based on sound and informed decisions. Now before you trust your life with it, you may want to think about breaking it in.
What is a break in period?
Breaking-in a new handgun is a simple range session (or series of sessions) where the firearm is put through its paces, confirming for you that it works without defects. This can be as little as 200 rounds or as many as 500 rounds of ammunition shot in sequences of controlled, aimed fire.
Do not fire all the rounds at once, but instead shoot 40-50 rounds, unload and lubricate (do not clean) and allow to cool before continuing to shoot. This will give you a good old-fashioned buildup of grime and gunk in the gun that will act as a lapping compound to wear in parts and experiment with stoppages and malfunctions. Once you have completed your break in session, thoroughly disassemble your firearm, and clean/lubricate.
It's best to start with a new and clean/properly lubricated firearm. If you are using a pre-owned but new-to-you handgun, have it checked by an armorer or gunsmith familiar with the design to be on the safe side. In either case, break it down according to manual before heading to the range and function check the handgun without any ammunition present to become familiar with its operation.
Almost all CNC-machined barrels in modern handguns leave the factory with microscopic burrs on the rifling and ramps. This is an unpleasant fact. These burrs generally will not affect the function of a firearm but if you shave them away with a few hundred rounds passing over them, the idea is that the rifling will have a better 'grip' on the bullet in the bore. Furthermore, if there is a defect with the firearm, be it an issue with parts fit or craftsmanship, the first few boxes of rounds will bring it to the surface and tell you whether the gun needs to be sent back to the factory (if under warranty) or turned over to your local smith (if not) to investigate further.
Even if you discount these theories, spending 200-500 rounds of practice time with the firearm is a good break in time for the user to become use to the operation, loading, unloading, and safety/decock levers, et al of the gun. Especially if you are going to depend on the gun for your self-defense.
Many firearms owners are not avid firearms shooters. While there is a large segment of the pro-2A community who regularly shoot at least once a month, an equally large segment can only make time to get to the range seasonally, or even yearly. I know one CCW-holder who has more than fifty firearms, most of which he has never shot even in a decade or more of ownership.
For some, it's not an issue of time so much as money. If you just saved up that $400 for a nice new imported 1911-clone and want to put 500 rounds through it at the range, well that's another $200 just for the ammo, plus the cost of targets, range fees, ear pro, eye pro, etc. etc.
Most modern firearms don't absolutely have to have a break-in period, but odds are you will see the magazines loosen enough to use them fully to capacity, the trigger smooth-out, and a slight increase in accuracy if you do. It's not for everyone due to cost and/or time constraints, but it is a wise idea. If there is a problem with the firearm, or a training issue you need to wrap your mind around to be able to use it, it's much better to find this out on the range than in real life.
Remember, paper men don't shoot back.