BitTorrent is a network and protocol used to share files, so BitTorrent itself cannot install adware on your computer.
However, the programs used to connect to the BitTorrent network and download files, called torrent managers or torrent clients, can and often do come with adware. The files you download can also contain malware and adware.
Stick to reputable torrent managers and, if prompted, refuse any offers to install additional software alongside them. These additional programs are often adware.
Likewise, be sure to only download and upload torrents you trust.