The kindly gentleman in the poem is the personified figure of Death. The speaker imagines him as driving along the road in a carriage. As she is unable to stop for Death—as the title of the poem clearly implies—Death must stop for her, which he does. Once he does so, he kindly invites the speaker aboard his carriage and proceeds to take her on a ride from this world to the next, slowly passing by the local school and fields of grain as the sun of the speaker's life begins to set.
Traditionally, Death was represented in works of fine art and literature as a quite frightening figure, a creepy skeleton or a faceless ghoul in a dark robe. But here, Dickinson presents him in a completely different light as a courteous, respectable gentleman riding in the kind of horse-drawn carriage that would've been a common sight on America's roads in the 19th century.
This is a deliberate choice on the poet's part. Far from seeing death as something to be afraid of, she presents it as nothing more than a journey from one world to the other. Death is being normalized here as a way of making it seem less scary or intimidating.