Buckner was an Old Army veteran and would do his duty no matter how difficult the particulars of it. The next morning Buckner wrote his old friend Grant a note asking what surrender terms he might get. Grant was with CF Smith at that moment, the two quickly hashed it out and Grant sent back his now famous reply, "No terms except unconditional and immediate surrender can be accepted. I propose to move immediately upon your works." Buckner had no choice but to surrender and his response was, "The distribution of the forces under my command, incident to an unexpected change of commanders, and the overwhelming force under your command, compel me, notwithstanding the brilliant success of the Confederate arms yesterday, to accept the ungenerous and unchivalrous terms which you propose."
Before this exchange of notes Floyd and Pillow worked hard to get as many men out of the fort as they could, mostly using arriving river boats to get their men out of there. Forrest also famously pulled his troops out, as well as about a 1000 other men who took the initiative to go along.
Grant was now a rising star in the Union. He would become a household name and people would say that US Grant stood for Unconditional Surrender Grant, not Ulysses S. Grant (actually born Hiram Ulysses Grant).
Up the Cumberland River is Nashville. The fall of Fort Donelson doomed Nashville and it would be abandoned by the Confederates in about two weeks.