We developed an approach to a quantitative characterization of entanglement properties of, possibly mixed, bi- and multipartite quantum states of arbitrary finite dimension. Particular emphasis was given to: 1) the derivation of reliable estimates which allow for an efficient evaluation of entanglement measures, 2) construction of measures of entanglement useful in the monitoring of the time evolution of multipartite correlations under incoherent environment coupling and experimental production of entangled states, 3) construction of quantities characterizing entanglement which are directly measurable (defined in terms of physically realizable operators). To this end we proposed generalizations of concurrence for multipartite quantum systems that can distinguish qualitatively distinct quantum correlations (generalized multipartite and multidimensional concurrences). We derived a lower bound for the concurrence of mixed quantum states valid in arbitrary dimensions. As a corollary, a weaker, purely algebraic estimate was found, which can be used to detect mixed entangled states with a positive partial transpose. We discussed also the monotonicity of the constructed quantities under local operations and classical communication (LOCC). We provided a condition for the monotonicity of generalized multipartite concurrences which qualifies them as legitimate entanglement measures. The constructed quantities can be, in principle, accessible in experiments to directly quantify the pure-state entanglement via a measurement of a physical observable.