CantThinkofaUserName,
When we read a passage of Scripture, it is often best to read the Scripture under the same heading (which are titles for different passages so that we can quickly identify them). This particular Psalm is under the title "God Is With Us" (according to the Good News Bible translation) and the first verse, which says "God is our shelter and strength, always ready to help in times of trouble. So we will not be afraid" sets the scene and purpose of the entire Psalm ... reassuring us that our God is a trustworthy God (v. 1,7) who is great and strong enough (v. 4-10) to save us from our problems. The whole idea of this passage is to reassure the believer that even if our world falls down around us, to God we can still hold on with a confident assurance that He'll carry the day.
When we fail to understand the 'atmosphere' or 'context' of a verse, we can be close to thinking something that is not right about God or the Christian life. Almost all of the arguments used by atheists can be brought down to them taking verses out of context in their zest to disprove Christianity and justify their faith that there is no God.
While I agree with the others in their interpretation of the passage (i.e. 'fear not' ... just think on who God is and know that He is mighty to save us from our troubles and fears and temptations), I do believe that we can become closer to God and perhaps experience His presence a lot more when we just withdraw ourselves away from everything and everyone just to be with Him. I often do this by going down to the beach and walk along it close to the waves or on the breakwater late in the night when you know that there'll be no one else around. Just taking in the scenery and talking and trusting that He's with me there, even though sometimes my feelings tell me that He's not.
Being still in this context is just sort of meditating or thinking about God and who He is in awe and the fact that He knows your name (just one in six billion people on one planet in a universe that is hundreds of millions of light-years in length). Just look around at everything and just think Wow, just who are you? If you can't see it, ask God to open your eyes to the beauty of everything and Him and believe that He will.
It isn't an inactive process, but an active process of taking time out from our lives to seek after God and just meditate (think about) on who God is, what He's done for us (in dying on the cross but also how He has provided for you and your family over the years), and even the simple fact that He knows your name, your sins, and yet still loves you.