LinkedIn members with whom you are connected. LinkedIn breaks these into three levels: 1st, 2nd, and 3rd, to indicate the nature of your relationship and how closely you are connected by other members.
A connection on LinkedIn is different than a Facebook 'friend' or a Twitter 'follower'. It is someone that you have a business relationship with and trust. You need to either invite these people to connect with you or likewise they invite you to do so.
There are several ways to invite someone to be a connection on LinkedIn. If you do a name search then LinkedIn will ask you to verify how you know the person and to provide an email address. It is possible to send an invite to a person if you worked with them or are in the same LinkedIn group.
Should you accept invites from everyone on LinkedIn? That is a personal choice, but realize that they will have your email address and access to contact you. You can remove people from your LInkedIn Connections by checking the boxes next to the names you want to remove. LinkedIn will not tell them that you have removed them from your list.