How to determine available free space on Ubuntu?

Yes, df -h (Disk Free) will show the free space on each of the mounted file systems.

So cd to the filesystem that's full, and du -sh * (Disk Usage) will show the total space used by each of the files/directories in the current working directory. The --max-depth option for du may also be useful here.

Finding exactly what is responsible for using all the space can be somewhat of an art - This answer lists some graphical utilities that can make that easier, though of course this isn't helpful in your case.

The simplest approach is just to work your way into the directory structure of the filesystem in question, trying to isolate files or directories that are taking up more space than expected.

Note: It's also worth running df -i to check the you haven't run out of inodes (ifree should be non-zero on writable partitions) - this can happen on some filesystems, especially if a larger number of small files have been created.

df to see space remaining. du to find what is using that space.