Many memorization or study techniques are also very helpful when memorizing the periodic table. However, the periodic table contains quite a bit of information. Generally, you will not need memorize the synthetic elements, which are the elements ranging from atomic number 104 to atomic number 118.
Repetition is the most effective way to commit information to memory. The Periodic Table is arranged so that elements within each vertical column are similar, or have something in common. These are called groups, and easily break down the period table for easier memorization. This process is called "chunking" and is effective as the brain remembers items better if you are able to connect them to one another.
Now that you have the sections grouped, you have some choices to make as to which memorization technique works out best for you. You may find a combination of techniques works as well:
1) Write It Out
Write out the elements by hand until you have them committed to memory. You should be writing out each group in order, and connecting each symbol with the name of the element.
2) Sing My Name
Say the name of the element and it's symbol out loud. You may want to sing this to make it easier to remember. You can sing the song in your head when writing a test or working on an assignment. Remember to create "songs" for each group. You could even connect them by singing "and then" between groups.
3) Create a Mnemonic
A mnemonic uses the first letter of each word to create a sentence, making it easier to remember. Like the song, you can repeat this in your head when writing a test or working on an assignment.
For example, the first family contains (in order) Hydrogen, Lithium, Natrium, Kalium, Rubidium, Caesium, and Francium. A sentence for this might be: Harry Little Nailed Karl Ruben's Cars Frequently. (Note that these are the Latin names, which makes it easier to memorize the Latin symbols)
4) Make Flashcards
If you wish to focus on memorizing the symbols of the elements more so than their placement, flashcards are a very good tool. Write the symbol for the element on one side, and the name of the element on the other. Show yourself the symbol and try to remember which element it is (check the back if you can't remember!)
This will train your brain to associate the visual of the symbol with the name of the element. Once you get good at this, you can look at the name of the element and try to recite the symbol. This challenges your brain!