Singularities exist in mathematical functions and appear when there is a division by zero. In particular classical theories which have the 1/r potentials theoretically have a singularity at r=0. Quantum mechanics turns these singularities either to fuzzy locuses, or forbids attaining the r=0 by the structure of the mathematical model, as happens with the solutions of Schrodinger equation even with a 1/r potential. When close to r, bound states take over, and discrete energy levels which forbid the r=0 ppoint.
The original Big Bang model was a classical model of General Relativity where singularities mathematically are allowed. In a similar way that a classical spherically symmetric explosion can be extrapolated to a point where all the mass is supposedly concentrated at r=0 ( a fictious theoretical point since in classical mechanics mass cannot be reduced to a point, it needs a volume), a General Relativiy model was built to fit the cosmological observations, and this model had a singularity at the beginning of time.
Before a time classified as a Planck time, 10^-43 seconds, all of the four fundamental forces are presumed to have been unified into one force. All matter, energy, space and time are presumed to have exploded outward from the original, General relativity singularity. Nothing is known of this period.
It was fairly successful originally, but then new observations demanded a quantum mechanical model for the very beginning of the Big Bang universe.
The current Big Bang model.:
Quantum mechanics had to enter because the homogeneity of the cosmic microwave background radiation cannot be modeled with classical thermodynamics, and an effective gravitational quantization theory for the beginning of time had to be used.
This also gives the quantum mechanical fuzziness at the beginning of the universe which diffuses the classical singularity at the origin.
What is the difference between a quantum fluctuation and a singularity in the beginning universe?
The quantum fluctuation sits/takes-over where the classical singularity would be, creating the fuzziness due to the quantum mechanical probabilistic uncertainty at that point. There are no infinities, as there are in the classical singularity.