We present an overview of diffusion models commonly used for quantifying the dynamics of intracellular particles (e.g. biomolecules) inside eukaryotic living cells. It is established that inference on the modes of mobility of molecules is central in cell biology since it reflects interactions between structures and determines functions of biomolecules in the cell. In that context, Brownian motion is a key component in short distance transportation (e.g. connectivity for signal transduction). Another dynamical process that has been heavily studied in the past decade is the motor-mediated transport (e.g. dynein, kinesin and myosin) of molecules. Primarily supported by actin filament and microtubule network, it ensures spatial organization and temporal synchronization in the intracellular mechanisms and structures. Nevertheless, the complexity of internal structures and molecular processes in the living cell influence the molecular dynamics and prevent the systematic application of pure Brownian or directed motion modeling. On the one hand, cytoskeleton density will hinder the free displacement of the particle, a phenomenon called subdiffusion. On the other hand, the cytoskeleton elasticity combined with thermal bending can contribute a phenomenon called superdiffusion. This paper discusses the basics of diffusion modes observed in eukariotic cells, by introducing the essential properties of these processes. Applications of diffusion models include protein trafficking and transport and membrane diffusion.