Is there a particular way to make Duas ?

Assalam akaykum. I see that in islam duas are very important but I don't how to make them! I mean is there a particular way? Do we just talk while joining our hands ? Can it be in our mind ? And when someone says make dua for so-and-so what do we do? How can we formulate the ask? Oh I'm sorry I just want to know how the best way to do it /:

Walaikum As’salaam,
 

I’ll try my best to answer your question, insha’Allah. c: Essentially, you can make dua however you wish, i.e. while joining your hands, in your mind, etc. :)
However, here are some dua etiquette as taught by the Prophet (Sallallahu Alaihi wasallam):

"Whilst the Messenger of Allaah (peace and blessings of Allaah be upon him) was sitting, a man came in and prayed and said, ‘O Allaah, forgive me and have mercy on me.’
The Messenger of Allaah (Sallallahu Alaihi wasallam) said, “You have been too hasty, O worshipper. When you have prayed and are sitting, praise Allaah as He deserves to be praised, and send blessings upon me, then call upon Him.”
— Al-Tirmidhi, 3476

The Prophet (Sallallahu Alaihi wasallam) said:

“Every du’aa’ is kept back until you send blessings upon the Prophet (peace and blessings of Allaah be upon him).”
— Saheeh al-Jaami’, 4399

The Prophet (Sallallahu Alaihi wasallam) said:

“The slave will receive a response so long as his du’aa’ does not involve sin or severing of family ties, and so long as he is not hasty.”
It was said, “What does being hasty mean?”
He said: “When he says, ‘I made du’aa’ and I made du’aa’, and I have not seen any response,’ and he gets frustrated and stops making du’aa’.”
— Sahih al-Bukahari, 6340 & Sahih Muslim, 2735.

The Prophet (Sallallahu Alaihi wasallam) said:

“No one of you should say, ‘O Allaah, forgive me if You wish, O Allaah, have mercy on me if You wish’; he should be firm in his asking, for Allaah cannot be compelled.”
— Sahih al-Bukhaari, 6339 & Sahih Muslim, 2679.

Also, here are some of the best times to make dua insha’Allah:
Prophet Muhammad (Sallallahu Alaihi wasallam) said:

"In the last third of every night our Rabb (Cherisher and Sustainer) (Allah (SWT)) descends to the lowermost heaven and says; ‘Who is calling Me, so that I may answer him? Who is asking Me so that may I grant him? Who is seeking forgiveness from Me so that I may forgive him?’”
— Sahih al-Bukhari (Hadith Qudsi)

Prophet Muhammad (Sallallahu Alaihi wasallam) said:

"There is at night an hour, no Muslim happens to be asking Allah any matter of this world or the Hereafter, except that he will be given it, and this (occurs) every night.”
— Sahih Muslim, #757

Prophet Muhammad (Sallallahu Alaihi wasallam) said:

"A supplication made between the Adhan and Iqama is not rejected.”
— Ahmad, abu Dawud #521, at-Tirmidhi #212, Sahih al-Jami #3408

Prophet Muhammad (Sallallahu Alaihi wasallam) said:

"There is an hour on Friday and if a Muslim gets it while offering Salat (prayer) and asks something from Allah (SWT), then Allah (SWT) will definitely meet his demand.”
— Sahih al-Bukhari

Prophet Muhammad (Sallallahu Alaihi wasallam) said:

"The nearest a slave can be to his Lord is when he is prostrating, so invoke (supplicate) Allah (SWT) much in it.”
— Sahih Muslim

I hope that helps insha’Allah! :) May Allah (SWT) bless you and your family, Ameen.