The valid relationship types and the meaning of the Index parameter are described for each.

RelationshipMember or LevelDescription
FirstMemberLevelBy default, returns the first member of the specified level. If the Index parameter is passed as x it gets the xth member of the level.
LastMemberLevelBy default, returns the last member of the specified level. If the Index parameter is passed as x it gets the xth member from the end of the level.
MembersLevelEquivalent to FirstMember or LastMember. If the index is positive it acts like FirstMember, if it's negative it acts like LastMember.
ParentMemberGets the parent of the member passed in. If index is 2, it'll return the grandparent (parent of the parent), 3 the great-grandparent etc.
ParentAtLevelMemberGets the parent or ancestor of the member passed in. If the index parameter is passed, this is the level number the parent must come from. Level number is one-based. (new in Version 8)
NextMemberGets the next member following the member passed in. If the index is greater than one, then it skips forward that number of members.
PreviousMemberGets the previous member preceding the member passed in. If the index is greater than one, then it skips backward that number of members.
FirstSiblingMemberGets the first member with the same parent as the member passed in. If the index is greater than one, then it gets the child at that index.
LastSiblingMemberGets the last member with the same parent as the member passed in. If the index is greater than one, then it gets the child at that index from the last sibling.
SiblingsMemberEquivalent to FirstSibling or LastSibling. If the index is positive, it acts like FirstSibling. If it's negative, it acts like LastSibling.
FirstChildMemberGets the first child of the member passed in. If the index is greater than one, then it gets the child of that index.
LastChildMemberGets the last child of the member passed in. If the index is greater than one, then it gets the child at that index from the last child.
ChildrenMemberEquivalent to FirstChild or LastChild. If the index is positive it acts like FirstChild. If it's negative, it acts like LastChild.
FirstDescendantAtLevelMemberReturns the first descendant of the member at a level specified by index. Index is one-based. 
LastDescendantAtLevelMemberGets the last descendant of the member at a level specified by index. Index is one-based. 
FirstDescendantAtDistanceMemberGets the first descendant of the member at a distance specified by index. An index of one is the equivalent of children. 
LastDescendantAtDistanceMemberGets the last descendant of the member at a distance specified by index. An index of one is the equivalent of children. 
DefaultMemberAnyGets the default member for the hierarchy. 
ItemLevelIf the Index parameter is passed as 'x' it gets the xth member of the level. Useful as it also supports sets using the MdxSet: syntax.