'However', 'nevertheless', and 'nonetheless' connect two sentences and show an unexpected result to what has been stated in the preceding sentence. 'However' is the most common of these expressions, whereas 'nevertheless' and 'nonetheless' are more formal are generally used in written English.
"However, the hotel stayed open despite the impassable roads."
"However, we decided to lower our operating cost."
"Nevertheless, if the multi-unit manager forgets to file he will be audited."
"Nevertheless, if we order take-out food we won't be able to stop working tonight."
"Nonetheless, the bankers are convinced they can succeed in the restaurant business."
"Nonetheless, we are going to the Food Costing forum."