As I know, standard ECDSA digital signature algorithm returns two values (according to this article or Wikipedia), but transaction push services or APIs, like this one, require one string as a signature. Can anyone explain, what is that string, and how can I get it with those values? Is it possible with OpenSSL? Thank you very much!
The ECDSA digital signature scheme returns two values. To be specific, the X and Y values computed on the elliptic curve are returned.
In Bitcoin the signture is DER encoded, which is represented as a string containing the X and Y values and also some header data. But both X and Y can easily be extracted from it when reading the string from left to right.
In fact the Bitcoin protocol uses more information than only those concerning the X and Y values. For example, the size of the DER encoded signature is specified, but also a so called signature type is used to define which part of the transaction the signature was computed for.
Take a look here under the section "Signing the transaction" to see exactly how it works. In the table you will see how a computed signature is used in a scriptSig.
The details regarding the DER encoding was discussed here.