Primordial nucleosynthesis a.k.a. Big Bang nucleosynthesis is theory of formation of first chemical elements in the early Universe. It is one of three evidences for Big Bang, along with expansion of the Universe and cosmic microwave background.

This theory explains how light elements, such as H, He, Li, as well as their radioactive isotopes, formed. All heavier elements formed a lot later in stars' interiors. 

Abundance of these elements is proof that the Universe was, once, hot and dense, and all conditions needed for thermonuclear reactions were meet.

During 50's leading theory of formation of elements was that all elements formed inside stars and during supernova explosions. This theory predicted that abundance of helium should be relatively small  if the only source is nuclear reactions inside stars, but the observation has shown that presence of helium is 25%, much more than it should be.
Second problem of this theory is deuterium. Deuterium is hydrogen's isotope, and it is actually hydrogen with one additional neutron in it's nucleus. The problem is that deuterium can't be formed in stars' interiors, but completely opposite, deuterium is being destroyed over there.
In today's modern theory of primordial nucleosynthesis we have matching of theoretical and observational values for all elements, expect for lithium, where we have unexpected significant deviation, and this detail is still huge challenge for this theory.

The process

Nucleosythesis started after the Big Bang, when the Universe cooled down, allowing deuterium to survive collisions with high-energy photons. At the beginning of the process proton/neutron ratio was 7/1. Free protons and neutrons are less stable than helium's  nucleus (helium's nucleus consists of two protons and two neutrons) so they tend to form it. But, formation of helium, requires some additional steps, and that is formation of deuterium.
The problem with deuterium is  that while the temperature is high enough, photons can destroy deuterium, and formation of  helium has to be delayed until the temperature is acceptable for deuterium.
When the temperature reached 1.6 GigaKelvins, sudden mass formation of elements happened. But, it didn't last for to long, because soon the Universe cooled down to temperatures that are not enough to support thermonuclear reactions.

At that moment, abundance of elements was fixed, and only fluctuations were because of radioactive decay of some elements.

The scheme of these processes is shown in the image below.

Let's explain this image. So, at the beginning, when proton(p) meets neutron(n), they form deuterium(D). Now, there are two possible steps. If deuterium (D) meets proton (p) 3-helium is formed, an helium's isotope (3He). Other step, if deuterium (D) meets another deuterium (D) then tritium is formed (T)  and one extra proton is released in this process.

3-Helium (3-He) and deuterium (D) together form helium (4-He) and one proton is released.

Also, helium (4-He) can be formed if tritium (T) meets deuterium (D).

Elements heavier than helium can be formed if 3-helium (3-He) meets helium (4-He), the result of this meeting is beryllium (7-Be), and if beryllium (7-Be) meets one neutron, lithium (7-Li) is formed.

Lithium (7-Li) can be formed also if tritium (T) meets helium (4-He).

If lithium (7-Li) meets one proton it splits into two heliums (4-He).

So these are all possible ways to form helium during the primordial nucleosynthesis.

As I mentioned before, deuterium played main role in this process. If too much deuterium is created, then there is no enough free neutrons to form heavier elements.  On the other hand, if there is no enough deuterium created, then we are missing one important link on our way to create helium.

Presence of helium should be huge, because helium is stable and all other elements end up forming helium. Possibility for helium to react with neutrons or protons to form heavier elements is extremely low.
Presence of lithium and beryllium should be extremely low compared to helium.

Elements

4-He

4-He means we are talking about helium, not about some of helium's isotopes. Number 4 means that there is 4 nucleons in it's nucleus, 2 protons and 2 neutrons. 

I mentioned before, that in the early Universe, proton/neutron ratio was 7/1. We need 2 protons and 2 neutrons to form helium, so we need 2 ''portions'' of nucleons for this reaction. Two ''portions'' contain 14 protons and 2 neutrons. 2 protons and 2 neutrons will form helium, and we will have 12 free protons. Free proton is equal to hydrogen's nucleus, because hydrogen's nucleus contains only 1 proton.

This is why helium's mass share is 25%, and helium number of elements share is 8%.

3-He

3 -He means that there are 3 nucleons in nucleus, 2 protons and 1 neutron, so 3-He is helium's isotope.

Primordial abundance of 3-He is not measured,because it was extremely low. It is assumed that stars of small mass can produce significant amount of 3-He

D

Deuterium is, we can say, completely opposition to helium, because helium is stable and deuterium can be easily destroyed. Primordial nucleosynthesis didn't transform all deuterium to helium.

Today, we don't know any process that can produce deuterium, because such a process should have temperature high enough to form deuterium, but not to high to start helium formation, and after couple of minutes, temperature should cool down to values which do not support nuclear reactions. Production of deuterium through the process of fission is not possible, either.

Be

Theory of primordial nucleosynthesis predicts amounts of primordial beryllium to low to measure.

Li

Lithium is the biggest problem of primordial nucleosynthesis theory, and here is why.

Lithium was produced as 7-lithium and 6-lithium. Observations of atmospheres of the oldest stars have shown that there is 3 times less 7-Lithium that it should be. Atmospheres of the oldest stars faithfully describe composition of the early Universe. In order to save the theory, some scientist have suggested a process in which 7-lithium is somehow transported into stars' interiors and destroyed over there. This process was never observed and still not fully theorized.

There is also problem with abundance of 6-lithium, because theory predicts that 6-lithium is created in amounts far below needed amounts to be measured, and scientist were able to measure amount of primordial 6-lithium, so obviously there is more 6-lithium than it should be.

Lithium problem is huge challenge for the theory up to date, but except for this the theory has shown amazing matching of theorized and observed values.