"V-neck" vs. "Round neck" of the dresses refers to the cut and shape of the neckline. V-neck dresses are cut to a point which (the neckline looks like a "V"), in a wedding dress, it is currently popular to be very low plunging to where the point ends somewhere along the sternum. Depending on how low it plunges will reveal more or less cleavage.
A round neck dress is softer than a V-neck - the neck is rounded or sometimes called a "scoop neck." Round neck dresses can be cut as high as the collar bone and go lower - the lower cut, again, the more cleavage is revealed.
"Mermaid" vs. "Trumpet" refers to silhouette of the dress, which means the shape, cut, and how it fits on the body. The mermaid and trumpet silhouettes are actually quite similar. They both are tight and form-fitting from the chest to about the top of the knees, then billow out at the very bottom - simulating either a mermaid tail or the horn of the trumpet. It seems the biggest difference is that in a "mermaid" cut the piece of fabric that forms the bodice and 'tail' actually ends (and usually has a hem-line) and other fabric is added to form the lower piece, or the 'fins.' Sometimes these are done with multiple layers.
A "trumpet" silhouette is created with all one piece of fabric that flows into the same shape but the line from top to bottom is never broken with a hem.
To describe in general terms, the difference between these two dresses is that the first one likely has much more angular lines. Fashion words used to describe it might include dramatic, geometric, or linear. The second dress would be considered softer (than the princess dress), elegant, and the piece would move from top to bottom without any lines to break it up.
Click the links below to see images of the closest thing I could find. The "trumpet" dress is not exactly round neck - but I wouldn't consider it a v-neck either.