As Large Language Models (LLMs) evolve into autonomous agents for long-horizon tasks, managing unbounded interaction trajectories under fixed context budgets becomes a core systems challenge. Unlike standard long-context documents, agent trajectories are heterogeneous and interleave observations, reasoning traces, and tool executions, so compression must preserve temporal dependencies, actionable state, and structural fidelity. Yet existing methods remain fragmented, making it difficult to compare design choices and reason about their reliability implications. This survey introduces a unified taxonomy of agent context compression along three dimensions: compression target (what is compressed), compression mechanism (how it is transformed and retained), and control policy (who decides when compression is triggered). We further organize recurring failures in compressed execution into F1: Pre-compression Decision Error, F2: In-compression Information Loss, and F3: Post-compression Access Failure, and examine domain-specific trade-offs in software engineering, web navigation, and deep research. By unifying the design space, failure taxonomy, and evaluation perspective, this survey provides a foundation for building scalable and recoverable LLM agents. A collection of papers available at https://github.com/YerbaPage/Awesome-Context-Compression.