Those words sound awfully fancy, don't they? It should be illegal to say them without a top hat and a monocle. They give two answers to the question, "So, what good are conditional statements, anyway?" They are used as a way of getting new information from information we already have.
The law of detachment allows you to "detach" the hypothesis from the conclusion. More precisely, if we know both p and p → q to be true, then we may conclude that q is true. When the traffic law says, "If the red light is blinking, then come to a full stop," and you see a blinking red light up ahead, it's clear what you'll do next (hint: resist the temptation to slam the gas pedal).
What can you infer from the following two statements: "All chickens hatch from eggs" and "Betsy is a chicken."
We can write the two statements in shorthand as follows: "chicken → hatch from egg" and "Betsy is a chicken." Since Betsy satisfies the hypothesis of chickenhood, she must by detachment meet the conclusion of egginess. In other words, "Betsy hatched from an egg."
The law of syllogism, on the other hand, allows us to squeeze together conditional statements. If we know both p → q and q → r to be true, we can squeeze them together to get p → r. After all, going through q when we can go straight from p to r would be just plain silly-gism.
"If you make a right turn, you must use a turn signal," and, "There is a right turn on the way to school." What can be inferred from these two statements?
As usual, it's a good idea to write the statements in their shorthand to make the structure of the implications clear. The first becomes "right turn → signal" and the second "school → right turn." (That is, if you go to school, then you must make a right turn.) Syllogism lets us write the chain "school → right turn → signal" and then cut out the middle part, leaving us with "school → signal." Translating out of our shorthand, "If you are driving to school, then you will use a turn signal."
We'll list the two laws one last time: