This is a quick overview of an Apex REST Callout using Basic Authentication from within Salesforce

Integration_BasicAuthRestCallout.cls

// Method to perform callouts
public String MakeCallout(string description, string keyToVerify){

    // define a response to caller
    String outcomeMsg;
    
    // define basic information for later, store these in a protected custom setting
    string endpoint = 'http://api.yourendpoint.com/'; // be sure this is configured in "Remote Site Settings"
    string resource = 'products/';
    string username = 'api_user'; 
    string password = 'passwordFromSettings';
    string method = 'GET';  
    
    // check to ensure a callout can be performed using the Limits class. 
    // This is 100 callouts per execution as of Spring 16 release
    if (Limits.getCallouts() >= Limits.getLimitCallouts()) {
        outcomeMsg = 'Maximum number of callouts has been reached.';
        
    // check for credentials error
    } else if (endpoint == null || username == null || password == null) {
        outcomeMsg = 'Please verify your API Credentials';
        
    // configure and perform the callout
    } else {
        
        // define transaction variables
        HttpRequest req = new HttpRequest();
        HttpResponse res = new HttpResponse();
        Http h = new Http();
        
        // Configure the request
        req.setEndpoint(endpoint + resource);
        req.setMethod(method);
        req.setTimeout(120000);
        
        // Add basic authentication to header
        // Create blob of user:pass
        Blob headerValue = Blob.valueOf(username + ':' + password);
        // Base 64 Encode the blob and prepend "Basic "
        String authorizationHeader = 'Basic ' + EncodingUtil.base64Encode(headerValue);
        // Add the basic auth string to the Request Header
        req.setHeader('Authorization', authorizationHeader);            

        // Configure standard headers
        req.setHeader('Accept', '*/*');
        // This tells the API that we are sending and receiving the data as a JSON object 
        req.setHeader('Content-Type', 'application/json');
        // Additional headers may be needed / Refer to the API documentation. 
        // Use a service like runscope.com to test everything ahead of time.
        
        // Set the body json with the description parameter, basically a string with a key value pair construction.
        // This will look very different for each integration resource.  
        // Some APIs don't use a body to take the request, 
        // they may simply take additional resources "/resource/order/Ord#" in the URI 
        // or parameters "resource/?orderId=133" in the URI
        req.setBody('{"text" : "' + description + '"}');
        
        // Attempt the callout - create return error on exception
        try {
            
            // Perform callout and set response
            res = h.send(req);

            // check response 
            if ((res.getStatusCode() == 200 || res.getStatusCode() == 201) && res.getBody() != null && res.getBody() != null) {

              // Deserialize the response untyped
              Map<String, Object> untypedMap = (Map<String, Object>)JSON.deserializeUntyped(res.getBody());
              
              // Check success of deserialization
              if(untypedMap.containsKey(keyToVerify)){
                    // The deserialized response contains the expected key!
                    outcomeMsg = 'Success!';

                    // ---------------------------
                    // ---------------------------
                    // Insert business logic here
                    // ---------------------------
                    // ---------------------------
                
                } else {
                    outcomeMsg = 'Error: Verify key not found in response';
                }
            } else {
              // callout failed
                outcomeMsg = 'Error: Callout failed. Please review the debug log for additional details.';
            }

        } catch (exception e) {
            // Unexpected exceptions will be caught here, like a deserialization error.
            outcomeMsg = 'Error: An exception has been encountered while calling out to Integration:  ' + e.getMessage();
        }
    }
    
    // Return the response
    return outcomeMsg;
}