1. Wrong: Peyton and Cloudy, two cute felines on campus.
Better: Peyton and Cloudy, two cute felines on campus, ran across our yard.
Also better: We saw Peyton and Cloudy, two cute felines on campus.
Explanation: This group of words had no verb.
2. Wrong: We want to go to the under-the-lights game across town, we have already painted our faces blue.
Better: We want to go to the under-the-lights game across town; we have already painted our faces blue.
Also better: We want to go to the under-the-lights game across town, so we have already painted our faces blue.
Explanation: The original group of words had two complete sentences separated by a comma, an error called a comma splice.
3. Wrong: As soon as the Blackhawks win another Stanley Cup.
Better: As soon as the Blackhawks win another Stanley Cup, all Chicagoan will celebrate.
Also better: Duncan Keith will get the recognition he deserves as soon as the Blackhawks win another Stanley Cup.
Also better: The Blackhawks will win another Stanley Cup.
Explanation: The original group of words was a dependent clause.