I was reading the datasheet of MP1482 chip which is buck regulator, I have read some of terms in the datasheet I did not understand could any one explain it in simple world.
You haven't supplied a link to the datasheet, so I'll only answer in general.
Synchronous is a common term in switching power supplies, and is short for synchronous rectification. In a basic buck switcher, there is a diode from ground to the input of the inductor. That works, but adds a diode drop of extra voltage across the inductor to "discharge" it more quickly than the output voltage requires.
The synchronous trick is to add a FET across the diode, and turn the FET on when you know the diode is supposed to conduct. That decreases the voltage drop, making the whole converter more efficient. The downside is that things go bad fast if this FET is ever turned on when the diode would not otherwise conduct. This is why synchronous rectification is something you have to wake up to design, unless you just buy a chip with it already incorporated.
Current mode is the name for one of the various possible control schemes of a switching power supply. This is mostly irrelevant to you when just buying a chip.
There is a controller in the chip that tweaks something to maintain the output voltage at the desired level as the load changes. In a switcher, there are various things that can be tweaked, like the pulse width, pulse frequency, and others. Current mode control is a particular scheme that decides to end each pulse when the inductor current has built up to a specific level. This reference level is what the controller tweaks in order to keep the output voltage at the desired value.
As with all control schemes, this has advantages and disadvantages. Current mode control simplifies the compenstation network, but that requires too much math to explain in this answer.
In terms of rectifiers, synchronous rectification means that the diode (which turns on and off as needed by the external voltage) is replaced by an electronically controlled switch, usually a MOSFET. The MOSFET is controlled by a signal that needs to be synchronous to the AC voltage for the rectifier to operate correctly. The advantage is lower loss by avoiding the diode drop. In a buck regulator, the flyback diode between ground and the coil is replaced by a MOSFET in synchronous circuits. The term asynchronous converter is uncommon, you would typically call them non-synchronous converters.
Current mode means that the regulation of the output is obtained by controlling the peak coil current. Monitoring the momentary output voltage could be called a voltage mode converter, but such designs are uncommon. Also one could omit current measurement and directly control the duty cycle. Current-mode converters handle the transition from continuous operation to intermittent operation (in low load scenarios) better than duty-cycle controlled converters. Also current mode converters turn off the switch quickly if the coil saturates, as this causes a quick rise of the current. Timing based converters need to be explicitly designed to prevent saturation, and most likely need margin.
The differences between all four have to do with the regulator's internal layout, and design choices. Synchronous and Asynchronous refer to two specific layouts regarding the input. Voltage-mode and Current-mode refer to the layouts and regulation sensing parts of the design.
Two good app notes for this are Synchronous vs. Aynchronous Buck Regulators and Switching Power Supply Topology Voltage Mode vs. Current Mode
