Listening is one of the four main skills when it comes to learning a language (the others being Speaking, Reading, and Writing). You can develop your listening skills in many ways, such as listening to the radio, podcasts or songs, and watching movies or videos. I've dealt with all of those in separate articles, so here I am going to list some sites which are specifically designed to develop your listening abilities.
DailyStep English — Audio lessons at five levels. Try a free lesson here.
Easy Conversations for Beginners — Hundreds of easy conversations with slow audio.
English All Year — The site is very basic, but there are some interesting activities.
English Conversations — 1,500+ ESL/EFL conversations on 25 topics. With transcripts.
EnglishClub Listening — Stories, songs, poems, news, and much more besides.
English Listening Exercises — An interesting collection of video-based lessons with quizzes.
English Listening Lesson Library Online —2,500+ free lessons with audio or video, a script, interactive quiz and vocabulary support.
English Media Lab — Free interactive exercises to practice listening.
English Speak —Listen to dialogues at normal or slow speed. 100 lessons at three different levels. Read my review here.
ESL Listening Lessons — Lessons at four different levels with questions and gap-fill activities. (see screenshot below). Part of the excellent TalkEnglish.com site.
Learn English Teens — Listening skills practice activities from the British Council, with worksheets and transcripts.
Randall's ESL Cyber Listening Lab — One of the oldest listening sites on the web. Quizzes at three different levels.
How to Improve Your English Listening — By Seonaid from Perfect English Grammar.
How to Improve Your Listening Comprehension — From PhraseMix.
How to Quickly Improve Your English Listening Skills Anywhere — From FluentU.
How to Really Improve Your English Listening Skills — From EngFluent.
Improving Listening Skills — From ThoughtCo.