If someone accidentally adds dots to your address when emailing you, you'll still get that email. For example, if your email is johnsmith@gmail.com, you own all dotted versions of your address:

john.smith@gmail.com
jo.hn.sm.ith@gmail.com
j.o.h.n.s.m.i.t.h@gmail.com
Note: If you use Gmail through work, school, or other organization (like yourdomain.com or yourschool.edu), dots do change your address. To change the dots in your username, contact your admin.

No one else gets your emails

No one can take your username
Your Gmail address is unique. If anyone tries to create a Gmail account with a dotted version of your username, they'll get an error saying the username is already taken.

For example, if your address is johnsmith@gmail.com, no one can sign up for j.o.h.n.s.m.i.t.h@gmail.com.

No one sees your mail

Your account is still private and secure. Emails sent to any dotted version of your address will only go to you.

For example, johnsmith@gmail.com and j.o.h.n.s.m.i.t.h@gmail.com are the same address and go to one inbox.

What to do if you get someone else's mail

Adding dots doesn't change your address, so dots aren't why you got someone else's mail. Instead, the sender probably mistyped or forgot the correct address.

For example, if someone meant to email john.43.smith@gmail.com but typed john.smith@gmail.com, the message went to you because you own johnsmith@gmail.com.