There are three types of invitations in Guild Wars:
Invite to a Party
Inviting another player to join you is the way to form a party and add to its members. To do this, click on the player's character (or if you can't see him/her hold down 'Ctrl' and click on their name) and then click on the tiny '+' button that will appear next to their name in the Party List window.
To join an existing party, click on any member of the party and invite them. this will be translated into a Request to Join that will be sent to the leader of that party to approve or reject. When a person looking for group members announces "invite self" or "SI", it implies that interested parties may simply invite the leader so that he or she may simply accept your "Request to Join".
An invitation to a party can also be made by using the "/invite" command followed by the character name of the player you wish to invite.
Tip: A quick way to invite a member you are interested in off the chat window is to click on their name. This will switch you to the Whisper Channel. Instead of chatting with them, click on their name, then press 'Ctrl+C' (to copy the name) then click in the invite line in your party list and press 'Ctrl+V' (to paste the name) then hit the tiny '+' invite button.
Invite to a Guild
An invitation to join a guild must be made by either the Guild Leader or one of the officers of a guild. On the members tab (default) of the guild window (default key is G to open this window) for a leader or officer, there is a text field where a character name may be entered and two checkboxes. The default check-box is "Member" where the invited person is invited to join the guild. The other is "Guest". An invitation to be a guest allows the invited player to visit the Guild Hall for 8 hours and participate in Guild Battles with that guild.
Each invitation to join, either as a member or guest, costs the inviting player 100 gold upon submitting the invitation.
Invite to an Alliance
An invitation to join an alliance can only be made by the guild leader of the guild that is the leader of the alliance. This can be done from the Alliance tab in the Guild window. The guild you wish to invite must be aligned with the same faction as you and they must not already be a member of an alliance.
Each invitation costs 1,000 gold.