First thing is first!
In the Yumekui Merry canon there are two worlds, the Real World and the Dream World. The Real World is exactly the same as ours, 21st century living with normal human beings. The Dream World, however, is more complicated.
A Nightmare is a type of 'Dream Demon' born from the hearts of human beings (it's explained as the shards of their dreams left behind when they sleep), or basically a monster as Merry puts it. Nightmares try to break through into the real would by finding a suitable vessel, these are people who have hopes and dreams that the Nightmares can hold hostage. They work their way into their subconscious through their dreams, and the moment that weakness is exposed, they use it to take over their body. This is done by pulling the person into the space between the dream world and the real world whilst the person is asleep, or during a quite literal "daydream". In these dreams, humans still look and act like humans, they aren't capable of anything more than what they are in the real world. When they wake up they usually forget what happened, just as a normal dream.
However, if this person is targeted as a vessel, or has already become a vessel, then the dreams become lucid and they will remember them. To them it feels like they've been pulled into another world and time stands still in reality. The basic theory behind all of this is based on REM sleep. Yumeji is an exception to this, he can enter peoples dreams without being a vessel, or being a target, and he has no control over when he'll be pulled into one.
Why is he an exception? Because the canon threw out the plot twist of: "he may in fact be a Nightmare himself."
So what does this mean?
Yumeji's physical existence in Luceti is the same as how it is in a dream. This gives him the ability to use powers he would not be able to use in the real world (and are explained below). But it also means that he cannot invade your dreams. Sorry, you'll have to leave that to Paprika.
And these powers?
His first power is seemingly useless, but it actually becomes the reason he teams up with Merry in the first place. Yumeji can predict what you're going to dream by observing your aura. The color of your aura is interpreted as what you will dream about that night. For example, a red aura would imply an exciting dream, filled with energy and power. Yumeji's accuracy with this ability is stated to be about 30% at best, it's something he's been continuously working on for the past 10 years.
However! He is always 100% accurate when he predicts nightmares. This is because the aura is pitch black with no color.
His second ability is called the "Lucid Gadget" (he named it himself). Using this, he's able to create and destroy things in dream world at will, as well as tie each of the dream demon's domains together. His power has been likened to that of both Pharos and Merry, putting him up there with the most powerful Nightmares in the dream world. The best example of this ability, and his most common use for it, is when he summons weapons used by the previous Nightmares he's encountered for use in his own battles, such as John Doe's guillotine saw and Noir's chains. When he uses the power obtained from another Nightmare his right eye changes to the same pupil shape they have (Nightmares have strange pupils). Yumeji is technically able to create anything he can imagine, but he does state that he can't imagine something he hasn't seen clearly before.
Yumeji can't copy the weapons or abilities of the people he meets in Luceti, only the Nightmares he's met in the canon.
As a third addition! Yumeji also has accelerated healing. This is due to the strength of ones resolve being a source of power in the dream world. Yumeji's resolve is absolute, he never gives up. Therefore, he can take one hell of a lot of injuries. He'd have to be knocked out or killed to be stopped once he's decided on something.
LONG STORY SHORT
He can summon weapons out of hammerspace and predict your dreams. Simple enough :D