Convenience for our whole household. It's always on, draws very little power, and makes very little noise or heat. My Synology NAS will allow everything in the house to back up to it, including the "Time Machine" backup system on Apple's laptops. It then automatically backs up the parts I care about offsite to Amazon Glacier, which costs me just a few dollars a month. If the house gets destroyed, my data won't.
Not requiring my Windows PC to be booted up is a big advantage here at home, since I leave town for a week at a time, and I don't want to leave my giant tower running just to act as a file server for a number of reasons.
I'll give you a perfect example of where this came in handy: my wife's old laptop finally died a couple of weeks ago. I knew it was coming, and she'd been itching for a faster, lighter one anyway, so I said to go get one; the old one was past the point where we wanted to spend anything to fix it.
She brought the new computer home, connected it to the WiFi, and it saw the NAS on the network. All she had to do was click on it and enter the password, and all her data, programs, settings, passwords, etc., were set up on the new laptop without her having to do anything, and without me having to walk her through anything while I was busy at work. Backups to the NAS were happening hourly in the background, so nothing was lost at all. She was absolutely thrilled.
I use it as a VPN server when I need one. I use it to record my surveillance cameras outside the house. I've used it as a web and database server when the need arises. Nicer models than mine will transcode video on the fly for streaming. It'll handle Bittorrent and other file transfers. And much more.
To put it succinctly, it's a powerful linux server with an awesome RAID system, an easy GUI, automatic security updates, and no hassle. Truly, my only regret is that I only bought a 2-bay model (DS216j), but that's all the budget allowed at the time. I'd get another Synology in a heartbeat. It's done exactly what I bought it for, and so much more.