How do I build libcurl from source?

Your permissions here don't matter, the reason you are getting that error is that it can't find the configure file.

In this case curl includes a buildconf file to prepare this for you. I successfully compiled it by doing the following steps.

First I make sure I have everything I need to build curl:

sudo apt-get build-dep curl

And then I run the following commands to build it:

buildconf
./configure
make
sudo make install 

This will put the library in /usr/local/

Looking at the sources, it seems like libcurl can use both cmake and autotools. Cmake is easiest:

cmake .

to create the build files, then

make
sudo make install

like you're used to.

For reference, here's how to use autotools. It is unusual for a git checkout to contain the configure script. You should expect an autogen.sh script (which creates and calls configure for you), or, failing that:

autoreconf --install
./configure
make
sudo make install

Both ways seem to work for libcurl.

If you're on Ubuntu 13.04 you might need to

ln -s /usr/local/lib/libcurl.so /usr/lib/libcurl.so

so curl finds its way.