The object of the present paper is to study a 3-dimensional normal almost contact metric manifold admitting Ricci solitons and gradient Ricci solitons. At first we give an example of a 3-dimensional normal almost contact metric manifold with α,β = constant. We prove that a 3-dimensional normal almost contact metric manifold admitting a Ricci soliton with a potential vector field V collinear with the characteristic vector field ε, is η-Einstein provided α,β = constant. Also we show that an η-Einstein 3-dimensional normal almost contact metric manifold with α,β = constant and V = ε admits a Ricci soliton. Finally we prove that if in a 3-dimensional normal almost contact metric manifold with constant scalar curvature, g is a gradient Ricci soliton, then the manifold is either α-Kenmotsu or an Einstein manifold provided α,β = constant.