What is called a 'mercenary' today is more like what used to be called a privateer: someone who is not a member of a nation's official military, engaged in military action for profit, with the approval of a soverign nation, and almost always the same nation the privateer calls home. Examples today include Blackwater/Xe, who provided security for US companies doing business in post Hussein Iraq, and also provided the initial security units for the post Hussein Iraqi leaders. Not as glamorous as Sir Francis Drake, but still very profitable. An ex special services soldier can make upwards of $250k/year in that dangerous business, if they don't mind running the risk of being lynched and strung up on a bridge in Baghdad.
Today, these hired guns can also be found on ships in the Indian Ocean, providing security against Somali pirates. The leaders of some of the oil rich nations also employ ex western special services mercenaries for their personal guard detail.
In general, mercenaries today provide security and occasionally intelligence gathering, as opposed to being used as units on offensive missions in years past. In the case of the US, they also fill in necessary gaps in military capability caused by budget constraints. For example, the security men who died in the Benghazi attack were ex SEAL/Delta soldiers, hired on to provide diplomatic security, as apparently there were no Marines available to fill that traditional role.
What used to be thought of as a mercenary was a freelance gun for hire, such as the ex WW2 soldiers who hired themselves out during the 1960's African wars that brewed up as the European nations abandoned their former colonies. Think Forsyth's 'dogs of war'. Back then, private companies were assembling small armies to attempt to seize control of mineral rich but cash poor African nations, seeking to install native 'leaders' with pre-signed mineral contracts in their hands. Those mercenaries would, as a general rule, be stripped of their citizenship in their native land for engaging in that activity.
One other form of mercenary: the French Foreign Legion, comprised entirely of foreign nationalities with French officers. The FFL would take just about anyone who demonstrates an ability to be a soldier and can pass the very rigorous training period, without looking too closely at their background.
French citizenship may be applied for after three years' service. Additionally, any soldier who becomes injured during a battle for France immediately becomes a French citizen under a provision known as "Français par le sang versé" ("French by spilled blood").
During the US Revolutionary War, the British, hard pressed by engagements elsewhere, hired Hessian mercenary units to take on some of the battles in the colonies. These were hired as entire units and not individuals, definitely a 'rent an army'.