Article II, Section 2 of the Constitution…
The President shall be Commander in Chief of the Army and Navy of the United States, and of the Militia of the several States, when called into the actual Service of the United States; he may require the Opinion, in writing, of the principal Officer in each of the executive Departments, upon any Subject relating to the Duties of their respective Offices, and he shall have Power to grant Reprieves and Pardons for Offences against the United States, except in Cases of Impeachment.
He shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur; and he shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law: but the Congress may by Law vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments.
The President shall have Power to fill up all Vacancies that may happen during the Recess of the Senate, by granting Commissions which shall expire at the End of their next Session.
So, in short, the President gets to nominate a candidate to fill a vacancy in the Supreme Court; but the candidate cannot actually be appointed to that position without the advice and consent of the Senate.
The current President, Barack Obama, certainly has the authority and the duty to make such a nomination at this point. The Senate is under no obligation to approve that nomination. I know that a few Senators have expressed the opinion that the next President should be the one to pick the person to fill the vacancy left by Mr. Scalia, and if a majority of the Senate agrees, then it is entirely within the Senate's authority to refuse to approve any new Justice nominated by the current President.