Compare:

Plot[Sum[i, {i, 1, n}], {n, 1, 50}]
Plot[Sum[i, {i, 1, Floor[n]}], {n, 1, 50}]
Plot[Evaluate@Sum[i, {i, 1, n}], {n, 1, 50}]

When you evaluate Plot[Sum[i, {i, 1, n}], {n, 1, 50}], floating points values of n are replaced into Sum[i, {i, 1, n}], giving something like, for example,

Sum[i, {i, 1, 20.4}]

The variable i then takes the values 1,2,3,... until 20 and then stops. This accounts for the steps in the plot. The parts in the plot without steps are probably where the refining algorithm does not feel the need to insert further points, because it did not detect a sharp curvature at its first attempt.

With Evaluate@Sum[i, {i, 1, n}] the sum is replaced by 1/2 n (1 + n), which gives a smooth curve.