With this new "sightings" feature, supposedly you see pokemon that are near you.
When they are no longer near, they should disappear from that list, returning when you move back in range.
Now, how can I use that information to find the pokemon in that list?
There's been a rather good post about how the sightings work on reddit, whereas I will summarize what's written there. Nevertheless, credits go to the OP 'Zakrael'.
For the following steps take this image, which is also contained in above mentioned reddit post.
Keep in mind thought that Pokemon will stay spawned for approx. 15 minutes, so if you take too long, the Pidgey might have despawned and therefore all the work has been useless.
Here is an alternative method that only uses some very easy high-school geometry. Let's say that you see the pokémon pop up in the sightings list at some point A below. (If you just happen to be somewhere inside of the spawn circle when you start, rather than at the edge, simply walk forward until the pokémon despawns and turn around)
Mark out where A lies irl in your mind or by placing some rock or the like. Then you are on the edge of the spawn circle.
Now, simply head straight forward until the pokémon disappears in the sightings. You are now at some point B at the edge of the circle. Mark out where this point B is in your mind or by placing some rock.
Try to slightly move to the left and right w.r.t. the line AB, just to figure out on which side of AB the pokémon appears in the sightings list. (This should take like a couple meters of pacing to both sides and checking the sightings list for the pokémon)
Once you know on which side of AB the pokémon lies, make a 90° turn from point B w.r.t. the line AB in that direction and walk straight until the pokémon disappears again. You are now at a point C. The line AC is then the hypothenuse of the right-angled triangle ABC, and ABC has all vertices lying on the circumference of the spawn circle. Therefore, by the inscribed-angle theorem, AC is the diameter of the spawn circle!
Simply walk from C towards A, and halfway there you will literally be on top of the pokémon.
Of course, it might happen that AB is the diameter itself, in which case you encounter the pokémon already in step 2, but it might be that your path reaching A is just a tangent to the spawn circle, in which case you can just make some pacing about in step 2, just like in step 3, to determine a good starting direction for the line AB, but that is sort of trivial for a sensible human being to do without further details.
Now, who says high-school geometry is without applications? :)
The easiest systematic method is walking in an outwards spiral pattern. This way, you reduce the issue of walking in the wrong direction and profit the most from starting near the Pokemon's location. The method is simple and is shown in the following picture:
There are 2 different scenarios shown here:
The optimal case is that when following your outwards spiral pattern (the path implies a street grid for added realism), you come within the spawn radius of the Pokemon at some point. This point is reached faster the closer your starting point is to the Pokemon.
In some cases, the Pokemon will disappear from your Sightings when you reach point D1. In this case, memorize the point on the map, shorten your spiral and continue as shown until you either encounter the Pokemon or it disappears again at point D2. Look at the map, determine the midpoint M between your current position (D2) and the point D1 you memorized. The Pokemon's location is roughly 200m from that point M in the direction most of your spiral walking path was. Determine that location on the map and go there to find it.
Another way to continue when you hit point D1 is to directly go 200m roughly in the direction of your starting point (and beyond it), then start in a spiral pattern from there again (unless you encountered the Pokemon on the way, of course). This only works in some cases, if you encounter D1 within the first 3 branches of the spiral, the method is very unreliable and you might walk in a completely wrong direction. Simply put, you should have completed a full circle before using this method.
I use Ingress, Niantic's previous AR game. If I'm in an area where I don't know the nearby spawn points, you can find some by opening Ingress. In Ingress, many Pokemon spawns are at the same location as XM, which are small clusters of lights. PokeStops and gyms likewise are in Ingress as larger, more complex structures with tons of lights around them. If you're in an area with many of those, this trick will not work, as the gyms and PokeStops will obscure the XM. In these cases, I fall back to the triangulation method that GrizZ outlines in his answer.
I try to avoid all that hassle by being prepared before going out to hunt Pokemon. If you're technically inclined (and don't mind breaking the TOS, with the understanding that it could get you banned), you can setup scanning software to scan the area to identify spawn points and what's spawning. Alternatively, you can search online for local Pokemon Go communities, as they will often have spawn points mapped, or at least where nests are located (nests are where specific Pokemon spawn often). I use these so I don't have to depend on the Sightings feature to capture the Pokemon I'm seeking.
When something pops up on the sightings simply do this:
If you are lucky, you will find it before it despawns, otherwise there isn't much else you can do to find it quicker besides having other people with you and have them do the same steps, but in different directions than you to cover more ground quickly.
If you see something in the sightings that you want to catch, walk around the area. The sightings only shows Pokemon within a 70m radius of your position, which is pretty easy to explore. If you can't find it, try closing the game and opening it up again — if the Pokemon disappears from the sightings, it wasn't there to begin with; sometimes they just get stuck on the display.
If you don't see anything exciting, try going somewhere else until you do! The purpose of the widget, and the game, is to inspire you to explore. Your best way to find Pokemon to catch is to cover ground, not to spend your time looking down and doing mental geometry.
