Why does my download speed drop when I’m uploading at the same time?

It is often the case that you cannot utilise your connection to its full capacity in both directions.
This is a characteristic of TCP, which is the protocol used for most internet applications.

TCP is a connection-oriented protocol, which means that it keeps track of whether the packets it sends arrive safely at the other end.
To do this, the protocol needs to ask the computer at the other end to send ‘acknowledgement packets’, known as ACK packets, once the packets have been received correctly.
If the sending computer does not receive these packets, it either stops sending or attempts to resend the packets it believes have been lost (depending on the situation).

This means that when you are fully utilising your upstream connection, you cannot send ACK packets to the site you are downloading from quickly enough, and so your downstream speed drops.

There are various ways to avoid this problem.
You can set your receive window higher (ask in an OS group), which means you receive more packets before sending an ACK. However, this has an unfortunate effect if you are downloading over slow connections.
You can prioritise these ACK packets so that they take precedence over smaller packets, but not all operating systems and routers support this.
Finally, you can configure the programme that is uploading (if it is an FTP client, for example) to use only, say, 80 per cent of the upstream bandwidth. This will leave room for the ACK packets.

The problem is most acute on asymmetric connections (because the upstream bandwidth is used up more quickly), but it occurs everywhere.