Many Amiga filetypes are different to the ones commonly encountered in the 21st century. The list below covers the vast majority of the downloads in this archive.
Disk images
Most of the titles for download are disk images, sector-by-sector copies of the actual floppy disks that changed hands.
DMS, short for Disk Masher, is the original Amiga-native disk imaging tool. It compressed by default, so that the output DMS file was smaller than the theoretically size of a disk. The vast majority of the disks in this archive were delivered in DMS format, so if you are looking for something as close as possible to the original, download the DMS file.
ADF, short for Amiga disk file, is a very simple copy of the disk sectors into a file. There are no headers and no compression, so each file has exactly the number of bytes of an Amiga floppy disk: 901,120 bytes, or 880 KiB. The vast majority of the disk titles in this format in this archive were converted from the DMS files.
ADZ is simply an ADF file that has been compressed using the gzip tool.
Package archives
Most of the packages on this site are packed into archives, a method of bundling lots of files together into a single file, compressed to reduce space. The archives are a transfer format - it's not normally possible to run software from an archive. Instead, once you have received a copy of the archive, you would extract it into a directory on your computer. From the extracted copy you could then do something useful with the files. You may be familiar with the Zip format used on Windows, or with tar+gzip on Linux and UNIX-like computers - this is the same principal.
Although zip does exist on the Amiga, you are far more likely to encounter LhA archives. LhA is also available for PC and Linux/UNIX, but the version commonly used on Amigas has extensions that properly preserve the Amiga's file protection bits.
You may also encounter LZX archives - this archive tool was newer than LhA, and used some tricks to get an improved compression ratio. It therefore became increasingly common when downloading from BBSes, where every minute you were online costed money, so every extra bit of compression was welcomed.