Use Can you put the... when you're talking to someone you know well, in a casual moment, and the thing you're asking about is obvious to both of you. Your housemate has just left the milk out after dinner, so you text them: Can you put the milk back in the fridge? Because you know each other and the milk and fridge are right there in both your minds, the direct form sounds friendly rather than blunt. With a boss or a new client, that same directness can come across as abrupt, so Could you put... is the better choice. The longer form adds a small layer of politeness that suits a relationship where you haven't yet built the kind of ease that makes a plain Can you feel natural.
When the wording fits the relationship
You're texting a housemate to drop something off. If the object's obvious to both of you, the direct wording feels friendly, not blunt: Can you put the milk back in the fridge?
You're at a friend's place with your hands full. Casual settings welcome the simple form: Can you put the keys on the counter?
You're leaving a note for family: Can you put the laundry away before dinner? It reads as a normal household ask, because they know you and the object is specific.
You're emailing a new client instead. Someone who doesn't know you yet has no history of friendly back-and-forth to soften the words, so the bare Can you put the files in the shared folder? can read as abrupt or even demanding. Adding a little distance with Could you put the files in the shared folder? signals courtesy and professionalism, and it usually lands better there.
Choosing the situation that matches your relationship
Adding please softens the request, and you can place it at the start, middle, or end: Can you please put the dishes in the sink? It's a recommendation, not a strict rule, so pick the spot that sounds natural to you.
After can, the verb always takes its simple form, so you'd say Can you put the towels away?, never Can you putting or Can you to put the towels away.
Quick check before you send: if you'd say the sentence out loud across the kitchen, can you put the fits. If you'd hesitate to say it to their face, like asking your manager to move a box or a stranger to slide over, Could you put the report on my desk? is the safer pick.