E stands for Electronic; Way means way only.
Do you brush your teeth twice a day? I do it with Dant Kanti.
Dant Kanti is produced by Patanjali from its factory at Haridwar.
And I live in Delhi.
To serve me in Delhi, Patanjali transports Dant Kanti from Haridwar to its dealer in Delhi.
The process is some what like this.
Patanjali issues invoice in the name of dealer in Delhi. And sends it to Delhi dealer by some medium. This looks normal as Patanjali is selling.
Patanjali has its transporters, who carry Dant Kanti from Haridwar to Delhi. Patanjali issues a tranporter bill to these transporters too. This transporter bill that Patanjali issues to transporter is generated electronically from GST website and could be used (and is valid too) only till the goods are on their way.
By combining e from electronic, way from transport way and bill from bill, GST Act named this transporter bill as EWAY bill.