Obviously it's very difficult to know what people may mean when they use the phrase "the leading of the Lord" but I think generally they are speaking of how the Holy Spirit is leading them toward a particular decision or response. And this is something God promises He will do for us, as seen in several passages such as the following:
I will instruct you and teach you in the way you should go; I will counsel you with my eye upon you.
As for how the Holy Spirit leads, that can be very different depending on the person. God doesn't lead us all the same way and it's impossible to nail down one universal method of detecting God's leading. (There are many in the Bible.)
For some people the leading of the Lord comes through Scripture. They pray and ask God to direct their steps, and then in the course of their Bible reading they come to a place where they feel the Lord is speaking through the Scriptures to bring specific direction.
Others depend on hearing something as the Word is being taught. I have had many people approach me over the years after teaching and tell me that something in the ministry of the Word spoke what they felt was a specific direction from the Lord for their lives.
Still others look for what the Bible calls the "still small voice" of God during their prayer time or just when they're listening for Him. (I want to be careful to add that although many Christians believe this method is more spiritual than the others, I don't believe that to be the case.)
The bottom line is that listening for God's leading is very challenging because usually we're pretty emotionally invested in whatever decisions we're facing, and that makes listening pretty hard. Especially if the Lord is telling us something we don't want to hear.
The point of all this is that you need to discover the way the Lord wants to speak to you. But there are no shortcuts to prayer and waiting on the Lord. And we must remember that God challenges all of us not to lean upon our own understanding. (Proverbs 3:5) That means, be careful not to do what you think is best but to really wait for God's wisdom.
When we're waiting on God it's often very helpful to employ some kind of fasting, and even to seek the prayer of others, asking them to intercede for you so that you might have "ears to hear."