making a map and calculate a distance

This is not an OpenStreetMap specific question.

In software, a formula called the Haversine formula is used to compute the distance between two points. You will likely find example implementations in your programming language of choice if you search for them on the web.

You can use a router e.g. an online one like yournavigation or OSRM, of course these can only be used for small amount of lookups (see OSRM api policy), and they candisapear at any moment. If you want to lookup lots of these distances you will have to install something of your own, installing Gosmore is quite easy, OSRM I've never tried to install.

These are the URLs to use.

http://yournavigation.org/api/dev/route.php?flat=59.31934&flon=18.0572&tlat=58.4162&tlon=15.5523&v=motorcar&fast=1&instructions=1
http://router.project-osrm.org/viaroute?z=6&loc=59.3193,18.0572&loc=58.4162,15.5523&alt=false

it's very simple to extract the distance from these results.

 OSRM: "route_summary":{"total_distance":199988
YOURS:  <kml><Document><distance>200.510334</distance>

See OSRM server API for more info

If you need to calculate a distance often some mapping software will help. I have recently installed an OSM map into Garmin's Basecamp and Mapsource both, or either can be downloaded free. They are very similar and I'm not sure why both exist or which is best. They're good and are great for route planning and sending maps,way points and routes (plotted or auto routing) to my Garmins along roads or paths.