There are two types of gifts in Mongolian tradition: "open-mouth gifts" and "closed-mouth gifts".
Open-mouth gifts are opened on top, like a cup, a bag, etc. Mongolians really like these gifts because they symbolise prosperity.
Closed-mouth gifts are opened in the bottom, like a hat, a lamp, etc. They're not offered very often because they symbolise debt.
A book is a very appreciated gift in Mongolia. When the recipient gets it, he/she brings it to the forehead to symbolise the knowledge this gift will bring him/her.
According to tradition, the most important gift for a Mongolian was the nine white gifts: eight white horses and and one white camel. Gold, silver, silk, clothes, khadags, dairy products and sweets were often added to this gift.
Another nomadic tradition was to offer the five domestic animals: camel, cow or yak, horse, sheep, and goat.
According to occasions and recipients, there are several sorts of gifts. For example, for a wedding, the most frequent gifts are furniture and home equipment. For the first hair cut ceremony, guests offer sweets, money ou toys.
If you ask someone to do a work for you, you'll have to offer him/her something. The gift will depend on the work done. There's also a tradition of gift when you visit a shaman for a ceremony, but the gift is rather a like an offering for the ceremony. It can be a khadag, dairy products, milk, money, alcohol, or food, depending to the type of ceremony.
When you offer a gift to a very respected person, like a monk for example, you must present it in a blue khadag. You put the khadag on your two hands and put the gift on it. Khadags are blue silk scarves that Mongolians like to offer.
When you visit a family, tradition wants you to offer a gift to thank it for hosting you. The present is made to the hostress at the end of the visit, before you leave. If you want to make a present to children, it's better to give it to their mother.
Nomads families always like to receive objects difficult to find in the countryside, such as batteries, cigarettes, matchsticks, sweets, headlamps...
When Mongolian people have guests at home, a gifts ceremony is held. They offer them alcohol, food (often mutton meat or cakes), and also exchange snuff-boxes, which can be considered as a gift.
The price of the gift is not important for the recipient, the single act of offering is important.
At the end, let's remind that tradition wants you to offer the gift with the right hand, and the left hand placed under the right elbow. It's the same thing when you receive the gift.