The Reasons Behind the Periodic Trends
To understand the specific periodic trends, we simply need to remember what we know about atomic structure and electron configuration. We can look at this in two peices: moving across the table and moving down the table.
Moving Across the Table
As you move from the left side of the table to the right (in a single row) what changes is the number of protons (represented by the atomic number) The further to the right you go, the more protons can be found in the nucleus. This makes the nuclear charge greater and the nucleus' attraction for electrons stronger.
It is also true that as you move from the left to the right on the periodic table, each atom also has more electrons. Since electrons are all negative, this would imply that each electron is increasingly repelled. It is even tempting to imagine that this additional repulsion balances out the additional attraction caused by the extra protons. Life (and atoms) however are not that simple.
As electrons are added in a single energy level they are placed in orbitals that are spatially oriented to avoid repulsion. As a specific example, Boron has 5 protons in the nucleus and 5 electrons. Carbon has an additional proton and an additional electron. All 6 of the electrons feel the extra attractive force of the extra proton. However, the new electron will go into an empty orbital. Assuming that the first p electron went into the px orbital, the next will either go into the py or the pz. These orbitals are oriented along different axes so that the repulsion gained from adding an electron is not as great as the attraction gained from the proton.
The result is that the addition proton matters a lot and the additional electron really doesn't matter much at all.
The simple summary
As you move to the right, there are more protons in the nucleus, so it is more positive and pulls harder on the electrons
Moving Down the Table
As you move down the table in a single column, or family, the number of protons increases as does the number of electrons. In addition, the size of the outermost orbital gets bigger; a 2s is bigger than a 1s, etc. Despite the additional electrons and protons, it is, in fact, only the size of the orbital that really matters when looking at periodic properties.
To understand why the number of protons and electrons is irrelevant, we need to relate the electron configuration of the elements to the properties themselves. Each of the five properties depends on the outermost electron, whether we are stealing it (ionization energy), adding it (electron affinity), pulling on it (electronegativity), or measuring the outer edge (both atomic and ionic radius).
The outermost electron is attracted by all of the protons in the nucleus and is repelled by the electrons on lower energy levels (as described above, the electrons on the same level largely ignore each other). If we think about a lithium atom the outermost electron (the third) is attracted by 3 protons and repelled by 2 electrons. So, as the outer electron “looks” toward the nucleus it “feels” the proportional pull of one proton (3 positive attractions – 2 negative repulsions = 1 positive attraction). This is called the effective nuclear charge. (There are, not surprisingly, some additional details, but this is a pretty good approximation and is certainly good enough for our purposes here.) If we then calculate the effective nuclear charge on sodium's outermost electron we find the same answer of +1. (11 positive attractive protons – 10 negative repulsive electrons = 1 positive attraction.)
Thus, the outer electron in sodium feels the same effective pull from the nucleus as the outer electron in lithium, but it is further away. Absence may make the heart grow fonder in poetry, but in chemistry distance makes the attraction weaker.
The simple summary
As you move down the table the electrons are further from the nucleus and, therefore, feel less attraction.