REST API - setting an activity's custom fields
You need to know the custom field ID of the field in question.  If the ID is 14, you can use:

&lt;www_path&gt;?entity=activity&amp;action=create&amp;activity_id=0&amp;target_contact_id=0&amp;custom_14=myvalue


In general, if you're asking these sorts of questions, you can answer them best by using the API explorer: Help menu > Developer > API Explorer.  You can use it to design your query, and it will build your REST string for you - including looking up custom field IDs.

To get the ID via the GUI, you can go to Administer menu > Customize Data and Screens > Custom Fields, then select "View and Edit Custom Fields" for the appropriate group.  If you hover your mouse over the "Edit Field" link, you'll see a URL that ends in "&amp;id=x".  "x" is your field ID.

To get the field ID via the API, do a call to CustomField.get.  Here's a REST call to get the ID of a field called "Mail Designation": 

/sites/all/modules/civicrm/extern/rest.php?entity=CustomField&amp;action=get&amp;json={"sequential":1,"label":"Mail Designation"}&amp;api_key=yourkey&amp;key=sitekey


Better still, use CustomField.getvalue to return just the ID:

/sites/all/modules/civicrm/extern/rest.php?entity=CustomField&amp;action=getvalue&amp;json={"sequential":1,"return":"id","label":"Mail Designation"}&amp;api_key=yourkey&amp;key=sitekey