“Elegy Written in a Country Churchyard” is about human mortality. This refers to the Latin phrase memento mori which means “remember your mortality.” The significance of this phrase, as it relates to the poem and your question, is that everyone dies. Death does not discriminate between the noble and the common. This poem is an homage to common people who led simple lives; that is, their lives are just as worthy of praise as the more famous, noble aristocrats.

In lines 17-20, the speaker notes that some commoners, rude forefathers, asleep in their graves, will never again awaken. "Rude," in this context, means uneducated which was common among the poor classes. 

The breezy call of incense-breathing Morn,
  The swallow twittering from the straw-built shed,
The cock's shrill clarion, or the echoing horn,
  No more shall rouse them from their lowly bed.

“Lowly bed” refers to the grave and to the bed each forefather slept in when he was alive. That actual bed is lowly because these were simple people; lowly meaning of low quality. Their homes and their beds were probably of low quality because they were poor.

The things which roused them from their lowly (poor) beds when they were alive (morning air, swallows singing, roosters crowing) will not rouse them from their lowly bed in the grave (low because it is six feet underground).