I am a mother who has a child out of wedlock who vaguely remembers her "biological father" when he was around. I repented from it and married someone else a few years later; he explained the opinions of the scholars regarding the child's ascription to me. My question is: do I have to tell my daughter that she was born out of wedlock? What if she asks what happened to him, and if so, when and how does one go about explaining it?

All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ʻalayhi wa sallam, is His slave and Messenger. 

You are not Islamically obliged to tell your daughter that she was born out of wedlock, and there is no benefit for her in doing so. Rather, it is totally against her interest.

If she happened to ask about her father, you are allowed to resort to tawriyah (saying something which has more than one meaning and intending a meaning different from what the listener is likely to understand). Verily, tawriyah is a legitimate way to avoid lying. You can tell her, for instance, that her father has passed away with the intention of meaning that a “legitimate” father does not exist. Allaah, The Exalted, referred to non-existence as death in the verse that reads (what means): {How can you disbelieve in Allaah when you were dead and He brought you to life; then He will cause you to die, then He will bring you (back) to life, and then to Him you will be returned.} [Quran 2:28] Ibn Katheer commented on the verse saying, “This means that He brought them from the state of non-existence to life...”

Please, note that some scholars held that the biological father has the right to give his name to his illegitimate child, and the child should be attributed to him in this case, which is contrary to the view held by the majority of scholars, as explained in fatwa 143587. We have also underlined in that fatwa that the illegitimate child may be given a general surname, like ʻAbdullaah or ʻAbd Ar-Rahmaan and so forth, in order to avoid embarrassment if he is attributed to his mother (indicating that he was born out of wedlock).

Allaah knows best.