Origin story
Saraswati is the goddess of knowledge, speech, music and learning, seated on a white lotus with a veena in her hands. As the river Saraswati faded from the earth, the goddess withdrew into the realm of speech and learning, becoming the patron of students, artists and scholars.
The veena is music and wisdom, the book is knowledge, the white lotus is purity, and the swan is the power to discriminate. Students begin their lessons with her name, and Vaishnava temples in Kerala keep a shrine for her beside the main sanctum.
Source: Wikipedia: Saraswati — https://en.wikipedia.org/wiki/Saraswati
Symbolism
The veena is music and wisdom, the book is knowledge, the white lotus is purity, and the swan is discrimination.
Stotra
2
Saraswati Gayatri
ॐ वाग्देव्यै विद्महे ब्रह्मपुत्र्यै धीमहि। तन्नो सरस्वती प्रचोदयात्॥
Oṃ Vāgdevyai Vidmahe Brahmaputryai Dhīmahi, Tanno Sarasvatī Pracodayāt
Meaning — May we know the goddess of speech, may we meditate on the daughter of Brahma. May Saraswati illumine our minds.
Recited for: Gayatri japa, clarity of thought
Saraswati Vandana
या कुन्देन्दुतुषारहारधवला या शुभ्रवस्त्रावृता। या वीणावरदण्डमण्डितकरा या श्वेतपद्मासना॥ या ब्रह्माच्युतशङ्करप्रभृतिभिर्देवैः सदा पूजिता। सा मां पातु सरस्वती भगवती निःशेषजाड्यापहा॥
Yā Kundendutuṣārahāradhavalā Yā Śubhravastrāvṛtā, Yā Vīṇāvaradaṇḍamaṇḍitakarā Yā Śvetapadmāsanā, Yā Brahmācyutaśaṅkaraprabhṛtibhir Devaiḥ Sadā Pūjitā, Sā Māṃ Pātu Sarasvatī Bhagavatī Niḥśeṣajāḍyāpahā
Meaning — She who is white as jasmine, moon and snow, robed in white, holding the veena, seated on a white lotus, worshipped by Brahma, Vishnu and Shiva: may that Saraswati protect me and remove all dullness.
Recited for: Prayer before study and examinations