docs: write API overview section

docs/API.md

@@ -18,7 +18,20 @@
 
 ## 1. Overview
 
-_TODO_
+Xpensely Server is a Spring Boot REST API that manages shared expense lists for pairs of users. It uses Google OAuth2 JWT tokens for authentication. All protected endpoints require a valid Bearer token in the `Authorization` header.
+
+**Base URL (local dev):** `http://localhost:8080`
+
+**Content-Type:** `application/json` for all request and response bodies.
+
+**Public endpoints (no auth required):**
+| Method | Path | Description |
+|--------|------|-------------|
+| GET | `/` | Health check — returns `"Welcome"` |
+| POST | `/api/users/createUser` | Register a new user |
+| GET | `/api/users/byName` | Look up a user by username |
+
+All other endpoints require authentication (see [Section 2](#2-authentication)).
 
 ## 2. Authentication