There are two parts here: the torrent file/link, and the downloaded contents.
The torrent itself is like a cloud-hosted website, cops and lawyers can see it and track who is using it but generally don't because that shit is expensive with so many individual sites and torrents. Using the torrent system itself is not illegal in the civilised world.
The second part is the actual content you're downloading. Doing so via torrent is not MORE illegal than any other way, but the ability to track your activity is different depending on how you do it.
And regarding the upload part, that might actually matter in some jurisdictions as you are supplying pirated material to others rather than merely using it in private (think dealing vs buying drugs). Torrent clients usually let you set rate limitations though, which you can potentially use to disable uploading entirely. Probably won't help you in a lawsuit and it means you aren't giving back to the community you're taking from, but blocking upstream does not have any immediate consequences in itself.
And as others have said remove the torrent file/link from the client once you're done downloading if you don't want to keep uploading long-term. Double-check the settings in your torrent client to make sure it does not automatically delete the downloaded content when you remove the torrent (this is not "necessary" but some clients might try for "convenience").