I’m trying download/generate WSDLs so that I can generate JAR files for use with my Java application, per the instructions here — https://developer.salesforce.com/page/Introduction_to_the_Force.com_Web_Services_Connector . The link says “To get your org-specific WSDLs, log into your organization and then click Your Name | Setup | App Setup | Develop | API.”. However, when I log into SalesForce, under my name there is only “My Profile,” “My Settings,” and “Log Out.” Where else do I need to look to get my XML-WSDL files?
As @Girbot pointed out, based on what layout you are in it is somewhat similar. In the new Lightning Experience, you can click on the gear in the upper right hand corner and select Setup Home. In Salesforce Classic, select setup next to your name in the upper right hand corner.
After both situations above you can then go to the quick find box in the upper right and type "API" where you will see options pop up.
In Lightning it shows: Custom Code --> API In Classic it shows: Develop --> API
As outlined by @sfdcfox, you are not a system administrator (more specifically do not have the permission "Customize Application") and therefore do not have access to the API option in the setup area. Please see the article Downloading Salesforce WSDLs and Client Authentication Certificates for the permissions needed.