'put' is usually translated as 放 fang4. So put on is 放(location)上. Similarly, put in would be 放(location)里. Your sentence about the glass jar would be: 把玻璃罐放桌子上. 'Take' is 拿 na2. Usually it's used with a resultative ending like 拿走 as in 他把玻璃罐拿走 （he took the glass jar and went off with it)，他拿掉(na2diao4)了玻璃罐 he removed the glass jar. Disclaimer - I'm not a native speaker so could be talking crap ;-/

1. to put a glass jar on a table =把一个玻璃杯放在桌子上。 2. to take it from a table=把它从桌子上拿走。

- To put :把 sth 放 +place+prep However , the 放always combine with anonther character to become a word. Like 放进(it means put in , and we always use the prep “里”)，放到（it means put on ,in or at ，and we always use the prep “里”“上”)，放在（it means put on ，inor at ，and we always use the prep “里”“上”)。 For example 1."to put a glass jar on the table" ="把玻璃缸放在（到）桌子上" put=把……放在…… 2.I asked my young sister to put her money in my purse. 我告诉我的妹妹把她的钱放进我的钱包里。 put =把……放进…… - remove sth to some place 把sth 拿到(放到，挪到，移到）+place+prep remove sth from some place 把sth 从some place +prep拿走（挪走，移走） For example, 1.Remove a glass jar to the table . 把玻璃缸拿到(放到，挪到，移到）桌子上。 2.Can you remove your bag from my chair . 你能把你的包包从我椅子上拿走吗？ Its a little difficult to recognize when is better to use “拿到“"放到”“挪到“”移到”or“拿走""挪走""移走".So I think its better for you to use "拿到，放到"and "拿走"when you are not sure，because they are frequently-used and can replace "挪到，移到"”挪走，移走“in a lot of cases . However , when the stuff that you want to put and remove is very heavy , you use "挪到，移到"”挪走，移走“，the native speaker will think your Chinese is really perfect due to your words using is accurate.