Is it appropriate to include my brother's son? I'm using an online service to create a family tree. I have already included my own information. Is it a common / best practice to include the son and the wife of one's half-brother (I think that is the term) since we share only a father?

It is your family tree - you can include anyone you like. It is perfectly acceptable and appropriate to include your half-brother's son and his family.

It appears from your tag that you are using Ancestry.com. If this is the case, then living people should be automatically hidden from public view. Only you and the people you let access the tree would be able to see living persons. If using another online service, you may want to review privacy settings to make sure no personal information about living people is shown.