Cnoidal wave solutions and soliton solutions in a one-dimensional ferromagnetic chain with anisotropy exchange interaction and single-ion anisotropy are obtained through travelling wave solution method and the influence of single-ion anisotropy on cnoidal waves and solitons is discussed. The results indicate that single-ion anisotropy makes the cnoidal waves and solitons easy to excite in an easy-plane ferromagnetic chain but difficult in an easy-axis one, and makes them more stable in the easy-axis ferromagnetic chain but less stable in the easy-plane one. The influence of single-ion anisotropy on soliton excitations in a one-dimensional isotropic ferromagnet is also discussed.