A writer bravely throws this scene under my red pen:
A man’s hand reached out from the smoky-plated mirror and caressed River’s cheek. The dark pink nails glowed against her pale skin. A rusted manacle chain dangled from his fist, clinking softly as the open cuff slapped around her arm. River was trapped like an animal trapped in a cage. Her lips trembled, but she would not cry. Crying would only prove she was weak. Instead, she would make him suffer, but how? A sudden impulse spurred her thoughts into actions and her hand darted to the free-swinging manacle and she slapped the open end of the shackles onto his wrist, clicking it shut. She heard a startled gasp escape from his mouth and felt the jerk on her hand as he instantly drew his arm away. Too late, he seemed to realize what she had done. His arm was shackled to River’s wrist.
You win the first crit by putting up an interesting image that caught my eye as I was copy/pasting all the submissions into Blogger. However: if he's caressing her cheek, how is he also holding a manacle chain in his fist? I would assume a caress is delivered with an open hand. Or is there a second hand involved? Second: she took the manacle away from him? Easily? Not real smart to hand somebody a manacle while you're reaching through a magic mirror, is it... but then you say that now he's shackled to her. So... now I'm confused about the shackle. Maybe there was a previous scene and your readers knew she was already half shackled, and why half of it would be left open for her to play with, but if there wasn't then you need to be clearer about the starting situation and how River managed this.
On the whole, a good start. The highlighted phrase is a bit of wordiness I think you don't need. How about: On impulse, her hand darted...
“You—!” he exclaimed, his voice between surprise and outrage. Again, he pulled his arm back, jerking against the chain, and River’s hand went with him. “What have you done? You loony girl. Do you think I do not have a key for these chains?"
She caught a glimpse of her reflection in the mirror, and immediately stopped her fluttering eyes. Unprepared for his response, she cocked her eyes upwards and spied a key hanging from a hook. She snatched the key and slipped it down the bodice of her dress. Immediately she felt her face grow hot and her mirror image proved her cheeks now flamed red.
“You are a mad woman!” Who said this?
The voice from the mirror was calm, deliberate, almost detached. “What possessed you to do something as childish as that? What were you thinking?”
River paused for a moment, struggling to sort her thoughts.
Is this really happening? No, it can’t be. Mirrors can’t talk. It must be a memory, but whose? Where am I in this memory?
If somebody remembers a mirror talking, then mirrors must be able to talk. Is there a reason that River wouldn't know that..? is it weird...?
She drew in a deep breath and tried to recall the time and place. Oh, yes. This wasn’t her memory. There was a convict onboard a ship. The poor girl had been sold into slavery but she escaped and River had been asked to erase the memory to protect the runaway slave. Thank goodness she remembered, but River had no choice but to follow along until the memory played itself out.
Highlight: the runaway slave, or River?
“I refuse to let you take me below this God-forsaken ship with the common criminals housed there. I do not belong there any more than you belong there. If I go, you go,” she said as she pinched her lips together.
The man lost his speech and River could tell he tried his best not to smile, as the reality of what she had done settled in. Half-comical. Half-dreadful. Completely crazy. As usual.
River glared into his eyes. She would make sure he did not win without a fight. Yanking her arm to her waist, the man fell completely out of the looking glass.
Just a thought: you could put the "River glared..." paragraph before she says "I refuse..." Also: how about a little description of this guy?
“Give me the key back,” he demanded, extending his free hand toward her, palm up. “Give me that key or I’ll rip that dress right off your body and retrieve it myself.”
That's very polite of him. She's a prisoner, sold into slavery (according to the memory) -- why's he being polite to her, and not immediately beating her down and ripping her clothes off?
River shook her head and stamped her foot on the floor. “I will not return the key peaceably. I refuse to go below with the other convicts. I do not belong there.” River said.
The man narrowed his eyes and gritted his teeth. He reached out and grabbed the eyelet of River’s collar, but as he yanked the material, River stepped back and he fell against the frame of the mirror. A pink light flashed, and suddenly, River stood alone.
The two hundred year old memory dissipated like fog on a cold night, as if written in the Memory Book with invisible ink.
The runaway slave was 200+ years old? Okay. I'm left wondering: does this mean River succeeded in erasing this memory? I hope you explain that ASAP. Because you said that she was just following along -- so everything she did is what the slave did? How does that erase a memory? Or was it just the last bit of struggle that broke the pattern?
On the whole, this was a decent scene. It could use a little more place-setting: I don't know if this happened in a prison, on the slave ship, or somewhere else. Sketching in the surroundings some more would clear that up.