I want to power a bo motor of 300 rpm with a 18560 3.7v li-ion battery but in the same circuit I want to have dc socket to charge the battery. But I don't know what kind of dc power adapter I shall use for this circuit. Please help.

 

Welcome to AAC.

You need to start with a little more information:
1) I assume you are using a single battery cell. Correct?
2) What is the capacity of the cell? Do you mean 18650? They vary in capacity.
3) Do you want to operate the device while the battery is charging?

Let's assume you mean the 18650 battery with 2000 mAh capacity. Now you need to design or buy a battery charger. They are often called battery management systems (BMS). You cannot simply attach a wall adapter to the battery without damaging the battery.

Once we get to that point, there are lots of options. I have used ones from Microchip and TI. The Microchip MCP7831/2 and MCP7833 are popular. Adafruit, Sparkfun, and others offer that chip in their devices. If you want a DIY, you can buy the chip and build your own. It will run off almost any 5V wall supply or from a USB port. Those chips do not allow use of the device while the battery is charging. At least they don't offer that option in their datasheets.

Here's a link to a datasheet for the MCP7831: https://cdn.sparkfun.com/assets/learn_tutorials/6/9/5/MCP73831/2.pdf

If you prefer TI , here is its selection tool:
https://www.ti.com/power-management...rger-ics/products.html#p1152=1;1&p1153max=0;1
Many of the TI chips say you can operate the device while charging the battery.

All of the above chips will work with a 5V wall supply that can produce 500 mA or more.