What you will need to move is the following:
- The CSR of the certificate, this is the block of encrypted text that was supplied when you first bought this. You need to Copy the whole file it will start with --BEGIN CERTIFICATE-- at the top and end eith --END CERTIFICATE-- at the bottom. If your old hosting companie allows you to download the file it will probable have www or domain.tld (insert your domain name there) .crt There could be upto 3 chain certificates or CA certs, you will need these also.
- The private key, this again will be a block of encrypted text and will (or should say) -- BEGIN RSA PRIVATE KEY-- at the top. Copy the whole file including the -- BEGIN to the --END and save this this, if you can download it will have the extension of .key
- THe server types must match though from old to new. So if your old provider had the site on a APACHE server, then your new one should be this as well. If not you may need to either download a different version of the cert, if this was provided or if may in the worst case have to buy a new on.