The idea of competitive and non-competitive antagonism can be a tricky one to understand.
An antagonist could be a medicine which binds to a receptor on a cell and blocks access to that receptor. This will prevent an agonist (i.e. one of the body’s own chemicals such as a hormone) from binding to that receptor, blocking its effect.
The antagonist can bind in two different ways, competitively or non-competitively:
a) Competitive Antagonism
The antagonist binds to the same receptor that the agonist wishes to use, effectively preventing the agonist from doing so.
It is competitive because both the agonist and the antagonist are competing for the same receptor.
The awesome artwork of Animal Care student, Sarah Mills, illustrates this point beautifully:

Our heroine in pink represents the agonist which has a desire to sit in the chair (bind to the receptor). The unco-operative green chap represents an antagonist who will not allow the agonist to access the receptor. Image: © Sarah Mills 2015
b) Non-Competitive Antagonism
In non-competitive antagonism, the antagonist and the agonist are not trying to bind to the same receptor. The antagonist will bind to a different part of the cell but, due to the antagonist’s shape, access to the receptor is still blocked. It is ‘non-competitive’ because the agonist and the antagonist are not competing for the same receptor.