1) ''Anxious'': it means ''worried and tense because of possible misfortune, danger, etc...'' / ''to be uneasy.''
* If you are 'anxious about' someone or something, you are 'worried about' them. (E.g: ''I was quite anxious about Mom'')
* If you are 'anxious to do' something, you 'want very much to do' it. (E.g.: ''We are very anxious to find out what really happened'')
* If you are 'anxious for' something, you 'want to have' it, or you 'want it to happen'. (E.g.: ''I am anxious for a promotion'')
* If you are 'anxious that' something happen, or 'anxious that' something should happen, you 'want it to happen very much'. (E.g.: ''My parents were anxious that I go to college'')
2) ''Anxious'' and ''Nervous'': Do not confuse 'anxious' with 'nervous'. If you are 'nervous', you are 'rather frightened about something that you are going to do or experience'.
E.g.: ''I began to get nervous about crossing roads.''
E.g.: ''Both actors were very nervous on the day of the performance.''
3) ''Anxious'' and ''Eager'': Firstly, you should know that the word 'eager' means 'having or showing a keen interest (in something) or intense desire/impatient expectancy (for something)'.
E.g.: ''He is always eager to win.''
E.g.: ''She was eager to find out the truth.''
*** NOTICE: The word 'anxious' has a long history of use as a synonym for 'eager', but some prefer that 'anxious' be used only to describe those who are 'worried' or 'uneasy', as in the sentence ''He's anxious about his upcoming surgery''. Although resistance to the use of 'anxious' to mean 'eager' is waning (it is permissible to use both in the same sense), writers should be aware that there are still those who frown upon using the word in situations where 'no anxiety' is present.
So, here is a hint for you: Keep in mind that ''anxiety'' is more of a worry (than anything else) and that ''eagerness'' is more of an intense desire/avidity (for something).
Hope that helps!!! :)