Estonian postal services are provided by Omniva. You'll find post offices in bigger supermarkets, but as many smaller offices have been closed, it is best to check their website for updated and accurate locations, along with opening and closing times. Mailboxes are orange in Estonia, and usually can be found around shopping centers.
It has become very popular in Estonia to send packages via parcel machines. There are several service providers for that, and they are usually placed next to each other inside or outside of the supermarkets. Outside of the Omniva machines, you could use Smartpost or DPD parcel. Just find the machine, enter the contact details where the package should go, pay, put the package in the machine, and that's it! When someone has mailed a package to you, via the parcel machine, you'll receive an SMS with a code to open your locker. It is a fast and easy way for online shopping, as well.