Competitive adsorption is when you have 2 or more sorbates in your solution/gas and they all adsorbe on the same adsorption site. The component which adsorbes faster (depending on the concentration and the kinetics itself) will be the pre-dominant species on the sorbens. Competitive adsorption should be consider if you want to remove a species selective from your solution/gas.
Non-Competitive adsorption is when you have 1 or more sorbates in your solution/gas and at least the same number of different types of adsorption sites. The sorbates will only adsorb on one particular site, where other components won't adsorb. However non-competitive adsorption for a mixture of 2 or more components is quite unusual and hence mostly occur when there is only 1 sorbate present.