Dmitri Mendeleev is credited with making the first periodic table that resembles the modern periodic table. His table ordered the elements by increasing atomic weight (we use atomic number today). He could see recurring trends, or periodicity, in the properties of the elements. His table could be used to predict the existence and characteristics of elements that hadn't been discovered.
When you look at the modern periodic table, you won't see gaps and spaces in the order of the elements. New elements aren't exactly discovered anymore. However, they can be made, using particle accelerators and nuclear reactions. A new element is made by adding a proton (or more than one) to a pre-existing element. This can be done by smashing protons into atoms or by colliding atoms with each other. The last few elements in the table will have numbers or names, depending on which table you use. All of the new elements are highly radioactive. It's difficult to prove that you have made a new element, because it decays so quickly.
How New Elements Are Named