The best free-to-play browser-based strategy game is a hard list to put together since while strategy games can be their own genre, usually "strategy" is a feature in games from MANY genres.
Playing chess online, playing Hearthstone, or trying out Forge of Empires are all strategy games but come from widely different genres.
Here are a few games to get you started in your search for popular browser-based strategy games that you can play for free:
1. Forge of Empires
2. Elvenar
3. The Settlers Online
4. Game of Thrones Winter is Coming
5. Travian