The subject of this article originates from sources that either the author or the Scott Pilgrim personnel have deemed as non-canonical. To find out about what is considered "canon" see SP:Canon.
This microchip is a device used by Gideon Graves as a mind control device used on Ramona to follow Gideon's commands whenever he's around (Breaking up with Scott, Telling Scott that she was still with Gideon, etc.) and keeps her docile during Gideon's first fight against Scott in the movie. It is implanted on the back of her head and glows green when active. As it is only seen in the film adaption, it is not canon.
As stated, it gives Gideon almost full control of Ramona (as the latter says, a way to get inside her head) whenever he's around her. How much control is unknown but what is known is that she is kept docile throughout her scenes with Gideon (She sat quietly only answering questions given to her by Gideon). She also could not do anything during Scott's fight with Gideon (When she first attempted to do so, she only grunted with struggle then touched the microchip before becoming docile again). Although the chip keeps her in check, she is shown to have some control, such as being shocked when Scott "died".
It is unknown how long Ramona has had the microchip, but it was the reason she deserted Scott and sought out Gideon after Roxy Ritcher was defeated and slain. When Gideon confronted Scott, the chip kept her quiet. When Scott arrived and fought Gideon, the chip prevented her from interfering, though she did express shock when Scott was apparently impaled and killed.
During the second round, Knives Chau kicked Ramona in the back of the head, disabling the chip. Ramona remained docile throughout much of Scott's second battle against Gideon in order to lure Gideon into a false sense of security. After Gideon appeared to have beaten Scott and Knives, Ramona approached him with a smile on her face. Gideon, thinking that Ramona was still under his control, let his guard down. Ramona then surprised him by kneeing him in the groin, causing him pain, and allowing Knives to take his pixel katana away.
Behind the scenes
There is also an interesting theory in the fandom community that Gideon used microchips to control the League of Evil X's. After all, one has to wonder why a 20-something like Matthew Patel, who only went out with Ramona "back in 7th Grade", would hate Scott to the point of wanting to fight him "to the death". Or why Lucas Lee was willing to sacrifice a highly successful acting career to fight Scott on the set of his latest movie (and eventually commit suicide). Or why Roxy, furious that Ramona stopped dating her, would exclaim "If Gideon can't have you then no one can!" The logical explanation is that Gideon implanted each League member with a microchip to "control the future of Ramona's love life", which would explain why each member is driven to destroy Scott.
As stated in the bonus fact subtitles in the movie, Ramona had the chip before the bar fight with Roxy Ritcher as it states she briefly touches it after telling him they should split (not as get out of here split but as in split-split) calling into question how long the chip has been there.
The chip acts as a counterpart to The Glow from the books. Both the chip and the Glow are used by Gideon to control or manipulate Ramona's actions.