Half Controlled and Full Controlled Rectifier-Why we go for full controlled while half controlled can control voltage?
A Half-controlled rectifier, when it comes to power circuits, is typically a bridge rectifier that uses two SCR's and two diodes for a single phase bridge, and 3 SCR's and 3 diodes for a three phase bridge.. This was common a long time ago, when SCR's were much more expensive than diodes.
By the same token, a full-controlled rectifier uses SCR's for all control elements; 4 in a single phase bridge, 6 in a three phase bridge.
Edit to add (per comment below): A half controlled rectifier (full bridge) can only control power flow in a single quadrant, the first. Full controlled rectifiers can control power in quadrants 1 and 2. More, if you add a second bridge for 4 quadrant control, you must have a fully controlled bridge, as one has to be completely turned off when the other is firing.
When half control is used, load current never reverses. If full control is used, sometimes thyristors are fired at 3rd quadrant and allows negative current pass through the load.
Full control system is used when supplying power to a motor which sometimes generates reverse currents.
You seem to be asking about full wave and half wave rectification. "Full controlled" and "half controlled" make no sense here.
Full wave rectifiers produce a DC peak each AC peak of either polarity, which is twice per AC cycle. This minimizes the time between the output capacitor getting charged up. It allows using a smaller capacitor for the same ripple, or less ripple for the same capacitor.
A bare half wave rectifier produces a DC peak only once per AC cycle. The advantage is that it's simpler and has one less diode drop in series with the DC. It can also use a common ground with the AC input.
If the AC is coming from a center-tapped transformer secondary, then two diodes can be used to make a full wave rectifier with only a single diode drop in series with the DC. Using four diodes, you can get ± DC supplies, each full wave rectified and with a single diode drop in series. The drawback is that the transformer is larger and more expensive.
Use the correct terms of half wave rectifier and full wave rectifier, and you can find lots more about them out there.