If your sink has sprung a leak where it meets the countertop, you can typically fix it yourself without touching the plumbing. When sinks leak around the edge, it's usually caused by a break in the caulk line or a lack of caulk. If the lip of your sink still sits flush with the countertop, like it's supposed to, you can simply recaulk the sink. If your sink is suddenly not meeting your countertop correctly, you'll need to remove the sink to discover the root of the problem before you can fix the leak.
Use a putty knife to scrape away the old caulk from around your sink, if present. Cut away the caulk from the sink using a utility knife if the caulk won't peel away. Do not scratch the surface of the sink or the counter with the knife.
Scrub the surface where you removed the caulk with dish-washing soap and water or an all-purpose kitchen cleaner and a bristled brush. Scrape up any tiny pieces of caulk which may have clung to the counter.
Let the area dry completely for at least 24 hours.
Cut the tip of the tube of silicone caulk off at an angle. Apply a steady, straight bead of caulk around the entire outside edge where your sink meets the counter. Smooth the surface as you work to keep lumps from forming. Use a damp rag to remove any caulk you get outside the area before it has a chance to set.
Let the caulk dry and harden overnight before using the sink.