We present a systematic study on the physical properties of EuFe$_{2-x}$Ni$_{x}$As$_{2}$ (0$leq$emph{x}$leq$0.2) by electrical resistivity, magnetic susceptibility and thermopower measurements. The undoped compound EuFe$_{2}$As$_{2}$ undergoes a spin-density-wave (SDW) transition associated with Fe moments at 195 K, followed by antiferromagnetic (AFM) ordering of Eu$^{2+}$ moments at 20 K. Ni doping at the Fe site simultaneously suppresses the SDW transition and AFM ordering of Eu$^{2+}$ moments. For $xgeq$0.06, the magnetic ordering of Eu$^{2+}$ moments evolves from antiferromagnetic to ferromagnetic (FM). The SDW transition is completely suppressed for $xgeq$0.16, however, no superconducting transition was observed down to 2 K. The possible origins of the AFM-to-FM transition and the absence of superconductivity in EuFe$_{2-x}$Ni$_{x}$As$_{2}$ system are discussed.