The word Apostle means "messenger." One who is sent is based on an incorrect view of this being a compound word. All major subsequent lexicons after Thayers disagree that apostollos means "one who is sent." All are sent to preach the Word, as is discussed in the Great Commission. DA Carson talks bout this issue (root fallacy) in his book Exegetical Fallacies. The Apostle is a messenger sent from God.
This topic of gifts is highly debated. Many deny the sign gifts, and many of those that do this assume apostleship is a sign gift. They lump it with tongues, prophecy, miracles and healing.
Scripture does not say these gifts have ceased. But, the lack of manifestation of some of these gifts in today's churches leads people to reason that they are not for today. Also, Paul's description of apostles as having seen Jesus Christ on earth feeds this belief.
Today, people calling themselves apostles, to me, is presumptuous. I know there are pastors that do this, and I do not deny that this gift is possible to manifest itself today. But, I feel that most instances of people calling themselves "apostles" are treating this like the office of "pastor." Apostleship is not described as a calling, and it can possibly be a temporary position (some say Barnabas and Timothy were temporarily given the gift of apostleship in Acts.) People are called to the pastorate. There is no discussion of being called to apostleship.
I urge you to read about this from different viewpoints to learn about this topic. Read all pertinent Scripture, and read differing views on this to see the interpretations of Scripture. If not, check with your pastor and talk to him about it. And, with everything, hold it up to all Scripture's statements on this issue.
In Him,
Dave