1. Literally, a small mat laid out in front of the door to one's house on which to wipe the dirt from one's shoes, typically with "Welcome" or some other greeting printed on it. After living in an apartment for so long, the first thing I wanted to buy when I moved into a proper house was a welcoming mat. You don't need to take your shoes off, just give them a good wipe on the welcoming mat.
2. A friendly, welcoming attitude or gesture. Sometimes used ironically or sarcastically. The town is rolling out the welcoming mat to competitors from all over the world for its annual triathlon this weekend. You're putting out the welcoming mat for cyber criminals if you post your personal details online.