The names of the packages for programs and libraries in Debian (and Raspbian) are standardized, and sometimes differ from the names the libraries give themselves or the tag or symbol names for them.
You should search for "curl" on the package system, and you will find the curl-related libraries, which might be what you want.
Packages ending with a "-dev" are the ones you usually need when you want to have something compiled, they will install the includes and other files needed for that.
apt-cache search curl
To find out all the packages with "curl" in the package name.
You are probably looking for one of these:
libcurl3 - easy-to-use client-side URL transfer library (OpenSSL flavour)
libcurl3-dbg - debugging symbols for libcurl (OpenSSL, GnuTLS and NSS flavours)
libcurl3-gnutls - easy-to-use client-side URL transfer library (GnuTLS flavour)
libcurl3-nss - easy-to-use client-side URL transfer library (NSS flavour)
libcurl4-gnutls-dev - development files and documentation for libcurl (GnuTLS flavour)
libcurl4-nss-dev - development files and documentation for libcurl (NSS flavour)
libcurl4-openssl-dev - development files and documentation for libcurl (OpenSSL flavour)
I would try to get the last one, the openssl-dev.
apt-get install libcurl4-openssl-dev
And then try to compile the software again. If that doesn't work, try to install the "libcurl3" as a second option.