(covers information from several alternate timelines)
A memorial service was a ceremony similar to a wake or funeral, where people would pay their respects to a fallen friend or comrade. It is also more traditionally held then a funeral, if the fate of the loved one might be in question and a body is not present to bury.
In the alternate reality, Captain James T. Kirk presided over a memorial service in San Francisco for those killed in 2259 by Khan Noonien Singh and Admiral Alexander Marcus. (Star Trek Into Darkness)
In 2268, Commander Spock held and presided over a memorial service for Captain James T. Kirk when it looked as though he might be dead aboard the USS Defiant as Kirk was lost in the interphase of Tholian space. Spock held this memorial service, despite the objections of Chief Medical Officer Leonard McCoy feeling that a funeral should be held. But Spock thought otherwise due to both no body yet being found and that he didn't want to assume full command of the USS Enterprise, although a bit later Spock and McCoy listened to Kirk's last orders tape -- even though both denied doing so when Kirk was saved and Spock was proven correct that Kirk was still alive. (TOS: "The Tholian Web")
After Miles O'Brien and Julian Bashir were believed to have been killed aboard a T'Lani munitions cruiser in 2370, Benjamin Sisko and Kira Nerys planned a memorial service to be held a day later. Ambassadors E'Tyshra and Sharat later told Commander Sisko a memorial service was to be held on T'Lani Prime. (DS9: "Armageddon Game")
In an alternate 2372, a memorial service was held on the promenade for Benjamin Sisko who was believed to have been killed, although he had actually suffered temporal displacement. Many people came forward to speak about him, including Major Kira. Jake Sisko did not, as he felt that "no matter what I said about him, I'd be leaving so much more out.. and that didn't seem right." (DS9: "The Visitor")
The crew and civilians of Deep Space 9 held a memorial service for Morn in 2374. As per Lurian custom, the guests brought food and drink to sustain Morn in the afterlife. (DS9: "Who Mourns for Morn?")