Answer:

Apostolic refers to the foundations of the Church.  The Orthodox Church has had an unbroken line of overseers (bishops) and priests that goes all the way back to the original Twelve Apostles.  By the grace of the Holy Spirit, this has helped preserve our doctrine, and truly, the gates of Hell cannot prevail against it.  For example, in the Orthodox Church we have no issues with things like same-sex marriages, because Apostolic doctrine says that homosexuality is an abomination.  However, this does not mean that we will condemn people who are afflicted with this lifestyle.  We will accept them in the Church, but we will also help them to get out of that lifestyle.  We understand that people in this predicament are in a great deal of pain, and need a lot of help.  This is the mercy that Christ desired.  We never condone such a sin, but we try to help the sinner.  Other Christian groups are usually unable to deal with such a dilemma, but since they are not orthodox, they often are dealing with it by altering their basic doctrines, and so, moving away from what Christ taught.  We view this as a grave tragedy and sin: a sin because it defies the King of the Church, Jesus Christ Himself, and a tragedy, because no one can be helped through anything else than the Truth.