Eavestrough and gutters are words that can easily get confused, although the two words are slightly different.
A trough is a channel use to direct water. An eaves is a section of the roof that hangs over the wall of the house, so that the wall is protected from rainwater. Therefore, an eavestrough is a channel that is connected to the eaves, taking the water away from the roof and usually towards a sewer or drainage system.
A gutter is a more general term. It could mean any place where runoff is stored or directed. So, an eavestrough is essentially a type of gutter.
It could be asked then, why is the term ‘gutter’ so common? When people talk about their houses, they often say they have gutters instead of eavestroughs. This likely comes down to the word gutter being shorter and easier to say.
The term ‘gutter’ has other uses. For example, it could refer to a gutter along the street or a dug up ditch to catch overflow water. The term ‘eavestrough’, however, specifically refers to a gutter built at the edge of a roof for directing water.
One reason for confusing between eavestroughs and gutters could be regional. In some regions, people usually say eavestroughs, and in other, people usually say gutters, even though they are talking about the same thing. Generally speaking, in the northern US people tend to say eavestrough, whereas in the southern US people tend to say gutters. So be aware that when you hear the other term being used, that person may be talking about the same thing as you!