What is the relationship between the base current and collector current of a bipolar junction transistor (BJT) ?

For example, the 2SC2712 has h_FE specifications as shown in Table. 1. 
In order to make this specification possible, the BJT makes the impurity concentration of the emitter (E) >> the impurity concentration of the base (B) > the impurity concentration of the collector (C) and makes the width of the base thinner. As a result, under the voltage conditions of the active region (forward bias between B and E, reverse bias between B and C), the junction between B and E is turned on, and carriers of high density E flow into B. The amount of carriers flowing in is determined by the barrier height lowered by the voltage between B and E (base current). In addition, since the base thickness of the carriers that have flowed in is thin, they flow into C without recombination with the carriers in B. This achieves high h_FE.

In practice, h_FE is not constant even in the above active region, and h_FE increases as the reverse bias between B and C increases. This is because the depletion layer between B and C widens as the reverse bias between B and C increases, narrowing the effective base. This is called the Early effect (base width modulation).
Also, h_FE becomes low in the saturation region where B and C are forward biased. In this region, both B and E and B and C are forward-biased, so they behave like resistors instead of the transistor behavior described above. At this time, excess carriers stay in the collector. These excess carriers lower the resistance of the collector layer, which has a low impurity concentration, and lower the saturation voltage V_CE(sat) between C and E. This is called the conductivity modulation effect. However, due to the presence of this excess carrier, time (accumulation time) is required during the transition from on to off.
As shown in Fig. 2, h_FE is affected by ambient temperature.