*All quotes are taken from The Norton Shakespeare, based on the Oxford Edition.
In Act I, Scene 1, it is revealed that Olivia has vowed celibacy for seven years while mourning the death of her brother: "The element itself till seven years' heat/Shall not behold her face at ample view,/But like a cloistress she wil veiled walk/And water once a day her chamber round/With eye-offending brine--all this to season/A brother's dead love, which she would keep fresh/And lasting in her sad remembrance." (I.i.25-31) Olivia is most likely using this extreme excuse to deter Orsino from pursuing her.
When Olivia meets Cesario (Viola), she decides that she can stop mourning for the loss of her brother because she has found a new love interest.
olivia stopped wearing a veil and stopped mourning for her brother because since she saw cesario/viola,she fell in love with her/him. she wanted to show her beauty to cesario/viola.
she wanted cesario to fall in love with her too on seeing her beauty. but in vain as cesario is actually viola,a lady! it's not possible for her to fall in love with the lady Olivia.
viola came to olivia to tell her how much Orsino love her. but olivia always refuses the duke's proposals. olivia love only cesarion/viola.Olivia even unveiled her face just for the sake of cesario/viola,to show him/her how much she has love him/her.