apt-fast is a shell script to accelerate apt-get downloads using Axel, a command line download accelerator. Based on apt-fast, some other scripts have been created and I though they are worth mentioning:
apt-proz - which is maintained in a PPA by Damar Riyadi (thanks for the tip!), who also maintains apt-fast in his repository -, uses ProZilla to handle the downloads and Damar says it's faster (at least for him) then apt-fast (using Axel). You can get apt-proz from this PPA: ppa:tldm217/tahutek.net, then to use it, simply replace "apt-get" with "apt-proz" when installing / upgrading a package.
apt-metalink allows you to download deb packages from several sources concurrently by using repository mirrors (you'll have to add them manually in the sources.list file). It uses Aria2 to download the files and it has an extra feature compared to apt-fast and apt-proz: it can check the hash of the downloaded files. For more info and usage, see apt-metalink @ Github.