The five periodic properties we will be studying are Ionization energy, electon affinity, electronegativity, atomic radius and ionic radius. Let's take them one at a time.
Ionization Energy is defined as the energy required to remove an electron from a neutral atom. This energy is sometimes measured in terms of the voltage necessary to remove the electron which is why it is sometimes called ionization potential (potential is the fancy physics term for voltage). This process results in a positive ion.
Electron Affintiy is the amount of energy released when an electron is added to a neutral atom, making a negative ion. If it is not immediately obvious why adding an electron should release energy, remember the definition of potential energy; potential energy involves an attractive force and a distance. When the electron enters the atom it is coming from far away and the distance to the nucleus is getting smaller. As the distance gets smaller, the potential energy gets smaller. Since energy can't be destroyed, it must be leaving. This is essentially the same explanation that Bohr gave for how the Hydrogen tube gave off light, only in a much more extreme version; here the electron is falling, not from an upper level of the atom, but from completely outside the atom.
Electronegativity is the force with which one atom pulls on the electrons of another atom in a covalent bond. Remember that a covalent bond is not a sharing of electrons, but rather a tug-of-war for electrons.
Atomic radius is the radius of an atom. This can be measured in various ways and is sometimes described as the covalent radius (half of the distance between two nuclei in a covalent bond). For our purposes, it is simply a measure of how big the atom is.
Ionic radius is the radius of the ion that is isoelectronic with the nearest noble gas. This means that as you go across the periodic table from the left to the right the ions start with positive charges and then in the middle of the table (around C in the second row) switch to negative ions. Ionic radius is simply a measure of the size of these ions. That means that on the second row (period) of the table we are talking about Li+1, Be+2, B+3, C+4 (or C-4), N-3, O-2, and F-1.