Since I think the colon is the easier to use of these two punctuation marks, I will begin with a brief explanation of the colon. There are two main uses for it. First, the colon introduces long and/or concluding explanations: this very sentence (which you can see is not capitalized) is an example of what I just mentioned!
The second use for it is to introduce a series, a list, or even a long quote in which quotation marks do not appear. The thing to remember when using the colon is to put it after a COMPLETED sentence. In other words, do NOT use it immediately after linking verb BE as in this example: “Three things I like are: pizza, tacos and pasta.” You must conclude the first sentence and express yourself like this: “Three things I like are these foods: pizza, tacos and pasta.” The same holds true with prepositions. Do NOT use it immediately after a preposition as in this example: “She is blessed with: good looks, charm and talent.” Instead, do it something like this: “She is blessed with three attributes: good looks, charm and talent.”
There are a few other things uses for colons.. They are to be used after salutations of business letters. Do NOT use a semi-colon after a salutation. For personal correspondence, the simple comma will do. And of course, the colon is used to denote the time (3:35) and verses of scripture (Mark 4: 2-1)
Now for the semi-colon, which has the effect of a pause which is stronger than that made by using a comma yet is not as strong as a period’s full pause. First, the semi-colon’s function is to link two closely related independent clauses which are not joined by conjunctions (and, or, but, etc.) Notice in my first example to follow, the semi-colon comes before an idea which either restates or explains further the idea in the first independent clause: “After ten years, I did not recognize him; even his mother did not know him.” Also notice how the second idea might be a contrast: “They say it is a wonderful activity; I find it ridiculous.”
There are a few other reasons for using the semi-colon, but I will finish up by discussing the most popular way to use it: to link independent clauses which have been joined by conjunctive adverbs. Conjunctive adverbs are more than a few! Here is a list of some of the most useful ones: MOREOVER, IN ADDITION, FURTHERMORE, FOR EXAMPLE, THEREFORE, CONSEQUENTLY, IN CONTRAST, ON THE OTHER HAND, and HOWEVER. There are two ways to punctuate pairs of sentences using these conjunctive adverbs. You can introduce the second sentence with a semi-colon or you can finish the first sentence with a period. In these two examples, both ways are acceptable: “You must fill out an application for the school; moreover, you must present the college with a personal letter.”/ “You must fill out an application for the school. Moreover, you must present the college with a personal letter.” For variety, you can put a conjunctive adverb word or phrase into the second sentence with commas around it because these types of adverbs are transitional: “You must fill out an application for the school. You must, moreover, present the college with a personal letter.”
If you have been having difficulty knowing exactly when to use either the colon or semi-colon, I hope this little explanation has cleared it up for you!