Most NASes use HDDs, and HDDs are really bad at handling multiple requests simultaneously. It takes a few milliseconds to move the read/write heads from one file to another, which is an eternity (can drop a 100 MB/s transfer down to 0.5 MB/s). So if your daytime backup is running and you try to pull data off the NAS, you will notice it's slower to respond. If the PC is left on overnight, it's usually better to run the backups overnight partly for this reason. (The other reason is so your files aren't backed up while you're in the middle of modifying them. An overnight backup usually gets the files after you've worked on them for the day.)
The network speed shouldn't be slowed, except for any bandwidth taken up by the backup. Most backup programs compress the data before backing it up, so the speed constraint is usually the compression. The network transfer usually plods along at 5-25 MB/s, leaving plenty of overhead for other network tasks (assuming you have Gigabit ethernet).
The PC will slow down somewhat to do the compression. But backup programs usually run at low priority, so any tasks you're doing on the PC should take priority over the backup (your program gets as much CPU time as it needs, and the backup runs in whatever CPU time remains).