The story of Lord Ganesha in the form of an octet

|| अथ गणेश-चरित-अष्टकम् ||

वेदेषु ब्रह्मणस्पतिं तं धीशं गणनायकम् | देवानामग्रणीभूतं पुरुहूतं पुरोहितम् |
धनुर्बाणधरं वीरं युधिजयप्रदायकम् | अथर्वशीर्षे यं स्तुत्यं ब्रह्मरूपं प्रकीर्तितम् |
गणकऋषिणादत्तं बीजंमन्त्रं च यस्य हि | तं नाथं सततं वन्दे पदभक्तिं च देहि मे || 1 ||

In the Vedas, He, the Lord of the intellect and the Ganas, is called Brahmanaspati. As Brahmanaspati, he is the foremost among the Devas, leading them from the front as well as their priest. He is the warrior carrying the bow and arrow conferring victory in battles. 

In the Atharvasheersha (Upanishad), he is praised as the very Brahman. Therein, Ganaka rishi has given us His beeja mantra (Om Gam Ganapataye namah)

Such as Lord I constantly pray to, May he confer on me devotion to Him
  
स्मृतौ  विनायकः ख्यातं विघ्नोत्पाद्यत्वकारणात् | ततो तस्य हि शान्त्यर्थं मातृभिस्सह पूजनम् |
कालक्रमात्तयोरैक्यात् विघ्नकर्तेत्यकीर्तितम् | कार्यविघ्नभयाल्लोकाः तं भजन्ति पुनःपुनः |
जयेतिहासं लिखितं यं व्यासवचनोपगम् |  तं नाथं सततं वन्दे पदभक्तिं च देहि मे || 2 ||

In the Smritis, he is called Vinayaka as he is considered the cause of obstacles. Hence various pacifying rites were performed for Him along with the seven Mothers (called Sapta mata). In course of time, this form got merged with the form of Brahmanaspati and He was called Vignhakarta (cause of obstacles). People started worshipping Him again and again to ensure obstacle-free performance of activities. He was the one who wrote the Mahabharata (known as Jaya) as dictated by Vyasa.

Such as Lord I constantly pray to, May he confer on me devotion to Him

पुराणकाले विख्यातं गाणपत्यमतमिदम् | गणेशमुद्गलादीनां पुराणानां हि शासनात् |
तेषामेव समाविष्टाः पूजा-विधि-कथानकाः | गणेश्वरस्य चत्वारावतारस्य च वर्णनं |
यं शंसन्ति भ्रुशुण्ड्यादि भक्तोत्तमसुहृज्जनाः | तं नाथं सततं वन्दे पदभक्तिं च देहि मे || 3 ||

In the Puranas, the tradition of worshipping Ganapati (known as Ganapatya) became popular primarily through the Ganesha and Mudgala puranas. In them, we find the most popular stories, rituals and forms of worship of Ganesha. Therein is also mentioned the four incarnations of Ganesha as well as the praises of great devotees like Brushundi.

Such as Lord I constantly pray to, May he confer on me devotion to Him

स्वजनाकाङ्क्षिता देवी स्वतनोः परिनिर्मितं | बालकं समरे सर्वभवगणादिपराजितम् |
अन्ते तु शूलिनो हस्ते शिरच्छेदमुपागतम् | सामजस्यशिरो यस्य देहे रुद्रेण संस्थितम् |
देवानां प्रथमं सर्व गणानां नायकं हि यम् |  तं नाथं सततं वन्दे पदभक्तिं च देहि मे || 4 ||

Goddess (Parvati) desirous of having someone her own created Him from her body. That lad defeated all the hordes of Shiva in the battle. In the end, He was beheaded by Shiva who later placed the head of an elephant and revived the lad. He was then accorded the status of the 'Foremost of the Gods'

Such as Lord I constantly pray to, May he confer on me devotion to Him
 
द्रविडेषु प्रसिद्धं तमव्वैयार्कालतः परम् | जगतः पित्रोः ज्येष्ठपुत्रेण विश्रुतं हि तम्  |
ज्ञानफलमवाप्यर्थं स्वपित्रौ परिभूषितम् | काकरूपेणनिर्मुक्तं घटजस्यघटात्नदी |
गजरूपेण संयुक्तं स्कन्दवल्ल्योः पुरा वने | तं नाथं सततं वन्दे पदभक्तिं च देहि मे || 5 ||

In the Dravida region (mostly modern day Tamil Nadu & Kerala), His worship became popular from the time of Avvaiyar. He is venerated as the the first child of the Universal parents (Shiva and Parvati). 

Below are stories associated with Ganesha that have been mentioned in the verse above

He circumambulated his parents to get the fruit of knowledge. He was the one who released the river (Kaveri) from the pitcher assuming the form of a crow. He was the one who brought Skanda (Muruga) and Valli together assuming the form of a wild elephant.

Such as Lord I constantly pray to, May he confer on me devotion to Him

कर्णाटके तु दासेन कीर्तितं परिपूजितम् | कर्णाटकजसंतेन हि मोरयागोसाविना |
महाराष्ट्र[प्रदेशे तु प्रबलीकृतकारणात् | गाणपत्यमतमत्रोत्फुल्लितानेकवर्षतः |
ज्ञानेश्वर्यारभेऽपि यं संतश्रेष्ठं प्रपूजितम् |  तं नाथं सततं वन्दे पदभक्तिं च देहि मे ||  6  ||  

In Karnataka, He was worshipped and praised by the Dasa (Purandaradasa) himself, It was the Karnataka origin saint Moraya Gosavi who popularised worship of Ganapati in Maharashtra which has now been flourishing for many years here. He is worshipped by all saints of Maharashtra starting from Sant Jnaneshwar onwards.

Such as Lord I constantly pray to, May he confer on me devotion to Him

अधिदैवं चतुर्थीणां तेशान्द्वे तु महार्थके | सङ्कटहरा मासीना भाद्रपदा तु वार्षिका | 
लड्डुकं कदली क्षीरं नारिकेलं जलान्वितम् | अपूपमोदसंयुक्तं गणेशाय निवेदितम् |
दूर्वाङ्कुर-रक्तपुष्प-रक्तवस्त्रेणालङ्कृतम् |  तं नाथं सततं वन्दे पदभक्तिं च देहि मे ||  7 ||

He the Lord worshipped on the Chaturthi day of every fortnight. Of these, two Chaturthis are significant - the monthly Sankatahara Chaturthi (occurring on the krishna paksha or waning fortnight of every month) and the annual chaturthi of Bhadrapada month (celebrated as Ganesh Chaturthi). He is offered various delicacies such as Laddus, bananas, milk, coconut water, Appam, Modaka etc. He is worshipped with Durva grass, red flowers and red garments.

Such as Lord I constantly pray to, May he confer on me devotion to Him

अपर्णाजमात्मरूपमिनायुतनिभरूपम् | ईशसुनुमुन्दुरस्थं ऊर्जस्वी ऋषिरोपमम् |
 एकदन्तमैश्वर्यदंतमोङ्कारस्वरूपिणम् | और्वसुरवन्द्यन्देवमक्षरैस्समलङ्कृतम् |
सुब्रह्मण्यकरार्चितपदाम्बोजं यं मोक्षदं |  तं नाथं सततं वन्दे पदभक्तिं च देहि मे ||  8 ||

He is the son of Aparna (Parvati), the form of Atman (self in each of us), shining like a million suns, Son of Isha (Shiva), seated on the mouse, Lustrous like a Rishi, Single tusked, Giver of all prosperity, form of Omkara, worshipped by the Brahmins, who is adorned by the letters (various poems/verses)

His feet are worshipped by Subrahmanya (also name of the author); He is giver of liberation.

Such as Lord I constantly pray to, May he confer on me devotion to Him