The MCU is a distinct universe from the comics, but draws abundantly from them.
Dr. Strange was created by Stan Lee and Steve Ditko. The latter's influences and sources of inspiration are the key to answer your question. Ditko wished to create a mystical character based on a mix of eastern esoteric and religious traditions.
For example, the Eye of Agamotto is strongly linked to the Snail Martys charm and amulet of Buddhism, especially in Nepali traditions. The fact that Strange and other dabblers in the occult often conjure powers from both good and evil sources, and use both black and white magic, is a display of influence from Taoism. And most explicitly, the setting where Strange first learns magic is Tibetan.
One traditional eastern practice that Ditko incorporated into the comics was the usage of mudras. From Wikipedia:
A mudra (muˈdrɑ); Sanskrit "seal", "mark", or "gesture"; Tibetan: ཕྱག་རྒྱ་, THL: chakgya) is a symbolic or ritual gesture in Hinduism and Buddhism. While some mudras involve the entire body, most are performed with the hands and fingers. A mudrā is a spiritual gesture and an energetic seal of authenticity employed in the iconography and spiritual practice of Indian religions.
The mudra most associated with Dr. Strange is the karana mudra:
The karana mudrā is the mudra which expels demons and removes obstacles such as sickness or negative thoughts.
This resounds with the powers of the Eye of Agamotto, which can dispel evil magic and reveal paths. Dr. Strange is also mostly immune to common diseases due to a mix of reasons.
It is made by raising the index and the little finger, and folding the other fingers. It is nearly the same as the Western "sign of the horns", the difference is that in the Karana mudra the thumb does not hold down the middle and ring finger.
He can often be seen using other gestures, for different effects. This has to do with different mudras having different meanings and purposes.
Since mudras have power of their own, we can imagine that they are as much a part of a spell or incantation as are words, thoughts, and material components. In some way, removing a mudra from a spell or replacing it with another mudra would be like removing or replacing a part from a machine - whether the spell will work, and how different its effects will be, will vary from spell to spell and from mudra to mudra.
If you wish to learn more about Steve Ditko and/or meddle with powers not comprehensible to mortals, a good suggested read is The Lesser Book of the Vishanti, by Catherine Yronwode.
All I have said above explains the usage of mudras for the comics... In the MCU, their usage is never elaborated, and it is quite possible that things will stay like that. In cinema, things such as hand gestures for magicians are supported mostly by the Rule of Cool, and screen time is precious - so there is more pressure to leave it at that than to include exposition should we ever have a second Dr. Strange movie.