To hold part of a meeting in private, there should be an agenda item to resolve to exclude the public for specified agenda items. A reason should be given, in general terms. Some possible reasons are to deal with employment matters, to discuss a legal dispute or because the matter cannot be discussed without breaching an obligation of confidentiality.
From a practical point of view, the agenda item to exclude the public usually immediately precedes the relevant items. If it is passed, members of the public can then be asked to leave. It is also practical to make the private matters the last agenda items so that the public don't have to wait around to attend later items.
Excluding the public should be avoided unless it is really necessary.