My mom told me that during Dhuhr prayer or any prayer I have to say Doorood Ibraheem after Atayat. Is this a must? And will my Salah be accepted if I don't say this?
After the last tashahhud in any salah you are supposed to recite Salawat (durood) and a dua before tasleem.
Reciting the durood here is a pillar and obligatory according to the Shafi'is and Hanbalis and would render your prayer invalid if you omit it. They interpret the command in the following verse to be especially applicable in salah:
إن الله وملائكته يصلون على النبي يا أيها الذين آمنوا صلوا عليه وسلموا تسليما
Indeed, Allah confers blessing upon the Prophet, and His angels [ask Him to do so]. O you who have believed, ask [Allah to confer] blessing upon him and ask [Allah to grant him] peace.
Similarly they rely on the following hadith and its versions, interpreting it to imply a commandment:
قالوا يا رسول الله أمرتنا أن نصلي عليك وأن نسلم عليك فأما السلام فقد عرفناه فكيف نصلي عليك قال " قولوا اللهم صل على محمد وآل محمد كما صليت على إبراهيم وبارك على محمد وآل محمد كما باركت على آل إبراهيم إنك حميد مجيد
We said or the people said: Messenger of Allah, you have commanded us to invoke blessing on you and to salute you. As regards salutation we have already learnt it (tashahhud). How should we invoke blessing? He said: Say: “O Allah, bless Muhammad and Muhammad’s family as Thou didst bless Abraham and Abraham’s family. O Allah, grant favours to Muhammad and Muhammad’s family as Thou didst grant favours to Abraham; Thou art indeed praiseworthy and glorious.
However according to the Malikis and Hanafis, reciting the durood after the last tashahhud is a sunnah. This means that your prayer will be valid if you omit it. However some of them have noted that if you omit it intentionally and regularly then despite the prayers being valid, you would earn sin for abandoning the sunnah.
As for the ruling on the dua, see Is it fard to recite dua ma’thur ( دعاء مأثور ) before finishing the prayer?