The actual encryption strength on a secure website connection enabled by a digital SSL certificate is determined by the level of encryption supported by the visitor's Browser and the Hosting Server.
Once a connection between the server and browser is established, and a public key encoded in the certificate is sent from the server to the browser, the browser needs to generate a symmetric key for encryption of the data that will be transmitted further.Here, the browser and the server both will compare their lists of available encryption algorithms(Cipher Suits) and choose the strongest one they both support for symmetric encryption. Nowadays, modern Browsers and Servers usually use three algorithms for this:
For an IIS webserver, you can force your webserver to use only 256-bit encryption. In the case of Unix-based servers, available ciphers usually depend on an openssl library.Hence, if your website is hosted on a server whose web server supports 256-bit encryption, such as Apache 2.X, then site visitors that use a Firefox browser will have their transactions with your site protected by 256-encryption. Thus with the right combination of Web browser and Web server, you can establish a 256-bit encrypted secure connection.
How would Browser know whether to generate a 256 Bit key or 128 Bit?
This depends on the cipher suite negotiation between the browser and the Server, not on the certificate installed on the server. When the browser initiates the connection with a Client, it sends a list of cipher suites it supports. The server then picks the one it wants and responds back.
For any secure website, click on the padlock sign if you see cipher suite AES_128_GCM ECDHE_ECDSA under the Connection tab.
That means both server and browser have negotiated to use the cipher suite AES_128_GCM. They will use 128 Bit Encryption with ECDHA_ECDSA key exchange.
Most Certification Authorities tend to put "256 Bits" on their websites because it looks good from a marketing point of view. It's not wrong, but it can be misleading for people who don't understand that it's how your server is set up and what your clients support that matters.