Transfering ArcGIS custom symbology to another machine

You could save the layer out as a layer file (this doesn't store data):

Saving a layer file

And then import that layer file as symbology:

Import symbology

Or, if you want the data, and symbology included in one package, you could save the symbolized layer as a layer package. This would probably be the easiest and best solution.

Saving a layer package (scroll down)

After you create the layer package, you just open it on the new computer and it will add the data and symbology you created.

If you want the symbols separately as a style reference then access the pc have intended symbols and go to "C:\Users\USER_NAME\AppData\Roaming\ESRI\Desktop10.1\ArcMap". There you will find USER_NAME.style. Copy this file and paste in the location "C:\Program Files (x86)\ArcGIS\Desktop10.1\Styles" of the new pc.

N.B. In fact, if .style file extension is changed to .mdb then it is an ms access databse where you can change the properties of symbols.

When you send someone a layer file you are assuming that the fonts to create that symbology are installed on their pc. If that person opens the layer file and sees wingdings instead of your symbology you will need to send them the fonts located on your pc at (C:/window/fonts); they will need to store the fonts in the same location on their pc to get them to show up correctly in ArcMap.