Bitcoin Core is the leading implementation of the software enabling users to interact with the Bitcoin network. It is not owned by any single business or organization but is instead updated and reviewed by a community of worldwide developers.
The software was initially released by Satoshi Nakamoto (under the name Bitcoin), only to be later renamed to Bitcoin Core to avoid any confusion.
By running the Bitcoin Core code, a user effectively acts as a node on the network. They can independently verify the validity of blocks received, as well as transactions sent by other users. This keeps miners in check and means that the user needs not trust anyone (such as a wallet provider) to display the correct view of the blockchain.
Bundled into the software is a wallet. Users can use this directly from within the application, or tether external wallets to their node to validate received transactions.
Users that frequently transact in Bitcoin should consider running a node to enjoy the various privacy and security benefits. A typical software wallet (one that does not interface with the user’s node) queries third-party servers for the user’s balance.
This practice can be of some concern, as it allows the server to tie the user’s balance to their IP address. The third party can reasonably infer that the user owns the public addresses they ask about.
In a similar vein, total reliance on a block explorer is dangerous from a transparency standpoint. As users have only a single point of reference for their balances, it is easy for the server to transmit false information to them.