OK, here are several translations of verse (33:50) that I have pulled from http://yaqb.org/ (Great website for studying the Quran, by the way). This is only one verse of many that makes clear the purpose of polygyny, who should practice it and why. Insya'Allah I will try to post these verses in the next few days or so.

Literal You, you the prophet, that We, We permitted/allowed for you your wives/spouses those who you gave their rewards (dowries), and what your right (hand) owned/possessed from what God bestowed upon you, and your paternal uncles' daughters, and your paternal aunts' daughters, and your maternal uncles' daughters, and your maternal aunts' daughters, who (F) emigrated with you, and a believing woman if she presented herself to the prophet, if the prophet wanted that He marries her, clearly/purely for you from other than the believers, We had known what We had commanded/imposed/stipulated on them in their wives and what their right (hands) owned/possessed, so that strain/blame/sin not be on you, and God was/is forgiving, merciful.
Free-Minds O prophet, We have made lawful for you the wives to whom you have already given their dowry, and the one who is committed to you by oath, as granted to you by God, and the daughters of your father's brothers, and the daughters of your father's sisters, and the daughters of your mother's brothers, and the daughters of your mother's sisters, of whom they have emigrated with you. Also, the believing woman who had decreed herself to the prophet, the prophet may marry her if he wishes, as a privilege given only to you and not to the believers. We have already decreed their rights in regard to their spouses and those who are still dependant. This is to spare you any hardship. God is Forgiver, Merciful.
Yusuf Ali O Prophet! We have made lawful to thee thy wives to whom thou hast paid their dowers; and those whom thy right hand possesses out of the prisoners of war whom Allah has assigned to thee; and daughters of thy paternal uncles and aunts, and daughters of thy maternal uncles and aunts, who migrated (from Makka) with thee; and any believing woman who dedicates her soul to the Prophet if the Prophet wishes to wed her;- this only for thee, and not for the Believers (at large); We know what We have appointed for them as to their wives and the captives whom their right hands possess;- in order that there should be no difficulty for thee. And Allah is Oft-Forgiving, Most Merciful.
Shakir O Prophet! surely We have made lawful to you your wives whom you have given their dowries, and those whom your right hand possesses out of those whom Allah has given to you as prisoners of war, and the daughters of your paternal uncles and the daughters of your paternal aunts, and the daughters of your maternal uncles and the daughters of your maternal aunts who fled with you; and a believing woman if she gave herself to the Prophet, if the Prophet desired to marry her-- specially for you, not for the (rest of) believers; We know what We have ordained for them concerning their wives and those whom their right hands possess in order that no blame may attach to you; and Allah is Forgiving, Merciful.
Pickthal O Prophet! Lo! We have made lawful unto thee thy wives unto whom thou hast paid their dowries, and those whom thy right hand possesseth of those whom Allah hath given thee as spoils of war, and the daughters of thine uncle on the father's side and the daughters of thine aunts on the father's side, and the daughters of thine uncle on the mother's side and the daughters of thine aunts on the mother's side who emigrated with thee, and a believing woman if she give herself unto the Prophet and the Prophet desire to ask her in marriage - a privilege for thee only, not for the (rest of) believers - We are Aware of that which We enjoined upon them concerning their wives and those whom their right hands possess - that thou mayst be free from blame, for Allah is ever Forgiving, Merciful.
Arberry O Prophet, We have made lawful for thee thy wives whom thou hast given their wages and what thy right hand owns, spoils of war that God has given thee, and the daughters of thy uncles paternal and aunts paternal, thy uncles maternal and aunts maternal, who have emigrated with thee, and any woman believer, if she give herself to the Prophet and if the Prophet desire to take her in marriage, for thee exclusively, apart from the believers -- We know what We have imposed upon them touching their wives and what their right hands own -- that there may be no fault in thee; God is All-forgiving, All-compassionate.
George Sale O prophet, We have allowed thee thy wives unto whom thou hast given their dower, and also the slaves which thy right hand possesseth, of the booty which God hath granted thee; and the daughters of thy uncle, and the daughters of thy aunts, both on thy father's side, and on thy mother's side, who have fled with thee from Mecca, and any other believing woman, if she give herself unto the prophet; in case the prophet desireth to take her to wife. This is a peculiar privilege granted unto thee, above the rest of the true believers. We know what we have ordained them concerning their wives, and the slaves whom their right hands possess: Lest it should be deemed a crime in thee to make use of the privilege granted thee; for God is gracious and merciful.

It seems pretty clear to me what Allah has permitted for the Prophet and not the male population as a whole. Perhaps I'm reading this verse all wrong, if so then someone please point it out for me. But what I understand from this verse is that only the Prophet has been given the privilege to marry any believing woman who approaches him. Does this mean a man can marry any believing woman, as long as he approaches them? No, Allah mentions in this verse (33:50) what has already been appointed for the rest of the believers. And that I believe relates back to verse (4:3).