It's easier than it looks
First, register on the OSM site. You need to be registered to make changes.
Updating maps goes in these stages.
1: Get your track off the GPS.
2: Upload your track to OpenStreetMap.
3: Draw over the track, which smooths out the GPS wobbles and makes sure things line up.
4: Give the things you've drawn 'tags' which tell OpenStreetMap what it is.
1: This is device-dependent; you'll be getting a GPX file out of it.
2: Go here -
OpenStreetMap | Login and upload your GPX. Wait a while (circa 20 minutes, sometimes much faster) for it to me processed. You will recieve an email when it's done.
3: From just about anywhere, click Edit. This launches the online editor in your browser. Choose "Edit live" unless you really want to work offline. To view GPS traces, click the little GPS icon in the bottom-left that looks like a phone or calculator.
OSM has 'ways' and 'points'. A road is one way, made up of yea-many points. A dual carriageway is two one-way ways; a cycle track might be a parallel way to a road. Pavements are usually not marked, they're assumed.
Say you start on a main road and want to draw a side-road that doesn't exist yet. Mouse over the main road and it will light up; click now, and you get a drawing tool with a trailing line. Click neatly along your trace.
To end editing, hit return or double-click. To abandon this edit, hit escape.
To add a point, either double-click or drag one of the points from the palette at the bottom of the editing window.
To enclose an area, finish by clicking on your start point.
Un-tagged ways don't appear on maps, so you can leave the work half-done.
Once you've drawn your ways, select one to tag it. To select a way, mouse over it and click once: it will highlight your selection in yellow.
For most common roads there is a preset. See the box with a car / houses / trees icon in the bottom-middle? That is the preset-o-matic. Car = roads, bike=bike stuff, tree=land use and so on. Click through to something credible, then click 'no preset' and choose from the list. This builds a list of standard tags.
Because this is wiki-land, you can make up tags on the fly, but the standard ones get drawn on maps
Commonly, for example, you'd have highway=residential for a residential road; highway=footway for a footpath; amenity=pub for a pub point, landuse=forest for maintained woods, and so on.
You'll want to add detail. For example, highway=secondary will come with a blank ref= tag for the reference; so you can add B123. All highways want names, not all get 'em. Names are optional for everything, but are the most useful optional tag.
Some of the tagging schemes are simple, some of them more involved (amenity=restaurant, name=Golden Palace, cuisine=chinese) and some, like seamarks, are downright byzantine - but the basic stuff is basic.
So, to add that residential side-road:
1: Hover on the main road and click to start off attached to the road.
2: Draw your trace nicely - smooth out wiggles the GPS introduces using your memory that this bit was straight and this bit was a nice curve.
3: Double-click the end to finish.
4: The way remains highlighted so you don't have to select it.
5: Click the preset icon until you get the car.
6: Choose Residential Road then add the name into the box that appears.
7: Click off the road to de-select it and your changes are automatically uploaded.
8: Add manual tags if you like.
You can always make changes at any time. Be bold! An iffy map is more useful than no map at all.
Advanced: If you're adding cycle ways parallel to main roads, remember to join them up at junctions. This makes bike routing work better.
