Too many writers let each paragraph stand on its own. Yes, a paragraph is a discrete unit, but it is connected to other units through transitions. Normally a transition occurs in the first sentence of the new paragraph.
Remember that while the paragraphs all accomplish discrete purposes, the reason they appear in a certain order is because of the particular relationships among those purposes. Make those relationships explicit for your reader.
The easiest way to connect paragraphs occurs when you have a list of topics to get through. After the first paragraph, start the next paragraph with "Second, ..." Note that this structure can get boring rather quickly. Most essays require a more interesting set of transitions from paragraph to paragraph--from point to point. Repeating a key word or idea from the previous paragraph is usually enough. (For instance, in this paragraph, the word "connect" in the first sentence is another way of saying "make the relationships explicit." In the previous paragraph, the point was to advise you to connect paragraphs with transitions. In this paragraph, the point is to advise you how to do so.)
Other web sites can provide lists of transition words that connect one paragraph to another such as Nevertheless, Even so, But, Moreover, Furthermore, and so on.