The most common way of celebrating this occasion is with something called a Burns Supper, a meal which progresses following certain traditions. Generally it starts with the Selkirk Grace, which is often attributed to Burns, but actually pre-dates him:
canna eat,
And some wad eat that want it;
But we hae meat, and we can eat,
And sae let the Lord be thankit
As mentioned, the Burns Supper is given to honour Robert Burns' birthday, but can also occur at other times during the year. I attended one in Edinburgh when I was there one August. It was the first time that I had eaten haggis; It tasted rather like heavily seasoned turkey dressing. Below I'm going to post a video of someone performing the Address To The Haggis: