How to make a web service callout in Salesforce?

If you are using the javascript button then you have to make that method webservice then it will allow you to make the Api callout to third party.

Salesforce provides a WSDL (Web Service Description Language) files. They are called “Enterprise WSDL” and “Partner WSDL”. A WSDL is an XML-document which contains a standardized description on how to communicate using a web service (the Salesforce API is exposed as a web service). The WSDL is used by developers to aid in the creation of Salesforce integration pieces. A typical process involves using the Development Environment (eg, Eclipse for Java, or Visual Studio for .Net) to consume the WSDL, and generate classes which are then referenced in the integration.

Enterprise WSDL:

This WSDL document is for customers who want to build an integration with their Salesforce organization only. It is strongly typed, which means that it contains objects and fields with specific data types, such as integer and string. Customers who use the enterprise WSDL document must download and re-consume it whenever their organization makes a change to its custom objects or fields or whenever they want to use a different version of the API. For the reasons outlined above, the Enterprise WSDL is intended primarily for Customers.

Partner WSDL:

This WSDL document is for customers, partners, and ISVs who want to build an integration that can work across multiple Salesforce organizations, regardless of their custom objects or fields. It is loosely typed, which means that you work with name-value pairs of field names and values instead of specific data types. The partner WSDL document only needs to be downloaded and consumed once per version of the API.