Steinbeck's suspense in Of Mice and Men arises from his adept handling of foreshadowing. In the opening chapter, the author reveals Lennie's propensity to touch soft things and the problems that that desire has caused George and him. Similarly, Carlson discusses how to kill a dog with a shot to the head and promotes the idea of killing something (someone) when it is no longer needed. The first example warns the reader that Lennie's problem is beginning to escalate and that it is just a matter of time before it causes more trouble for George and him; thus, the reader anticipates the next scene between Lennie and Curley's Wife. The scene with Carlson demonstrates what he and others on the ranch think of "useless" creatures and makes the reader wonder what or who is next for a bullet to the head.
While these examples of foreshadowing make for an incredibly interesting read, Steinbeck's most meaningful use of suspense in the novella is George and Lennie's dream to own a place of their own. Each chapter seems to bring the men just a little closer to the realization of their dream--Candy has money and offers to go in with them; George knows of a specific place that they can buy; George finds a confidant in Slim and begins to think that they might just be able to get the farm. Steinbeck elicits excitement for the men from his readers by making the dream seem oh-so-close, and then Lennie kills his puppy and Curley's Wife, and the dream is lost. While readers do wonder at the beginning of Chapter 6 what will happen to George and Lennie, the most significant suspense has been resolved so that Steinbeck's theme of the unattainable American Dream reverberates.
Suspense is an anticipatory feeling, right? It's the moment in a movie at which you wonder what is going to happen next or you are sitting on the edge of your seat nervous for a particular character.
Suspense is built throughout a story in the rising action and usually the moment of greatest suspense is right before the climax happens.
In this novel, I see suspense around Lennie at most every moment we experience him. We wonder if he's going to mess up. George certainly does. We see this when George won't let him talk, and then when he does talk it almost loses them their shot to work there. We see it with Curley first meeting and antagonizing Lennie, we see it when Lennie is alone with the dog, and then greatly when alone with Curley's wife. The fact that it's Curley's wife he is alone with makes it a double dose of suspense because Curley will get really mad when he figures out what happens.
From this question, I am assuming you are questioning what was the major conflict of the novel. While there are many different conflicts, as John Steinbeck intended it, one of the major conflicts that caused suspense was George's decision to kill Lenny.
When faced with the dilemma of dealing with Lenny's actions through Curly, who would be cruel and not understand the ignorance that goes along with Lenny's character, or to take Lenny's life into his own hands, George decides the latter would be best. He attempts to make the decision that would, in the end, be best for everyone. The characterization of Lenny allows the reader to love and empathize with Lenny, but the reader also comprehends the faults that Lenny's impairment ultimately comes with. This duality is what tears the reader's heart as we try to place ourselves into George's position -- What would a true friend do?