GPS and similar systems (Russian GLONASS, Chinese BeiDou, for example) are the first thing that comes to mind. Modern militaries rely heavily on extremely accurate positional systems for navigation, coordination, and intelligence gathering. The loss of GPS would be crippling for aircraft and ships, and extremely inconvenient for land forces. Drones and missiles may or may not be effected, since some use satellite navigation and communication and some do not.
How could they adapt? Land forces would be able to adapt most quickly, given that static maps, roads, rails, and landmarks are still there. The US Navy and Airforce, I'm told, still teach traditional navigation techniques so after some brushing up they could probably be operational again soon. I'd assume every major power does also, or would quickly learn.
General communications would be lost too. Terrestrial communications are dominant across domestic territory, ranging from optical fiber to point-to-point microwave transceivers. Across borders, in the air, and in the ocean, however, satellite is very important for maintaining communications. The ability for military forces to coordinate in real time would be devastated.
How could they adapt? During WWII, the allies maintained a network of radio repeater stations. Artillery units would send target information and wind conditions out over radio, the message would be repeated from station to station all the way back to England or the USA, mechanical computers would calculate the firing angles, and those angles would be sent back along the network. The turn-around time was only a few minutes.
Today, there's several independent networks of terrestrial radio repeaters run by various private and government organizations. These networks are usually used during disaster relief operations, but they could be co-opted for warfare easily enough. Modern consumer radio transceiver technology allows a hobbyist to reach clear across the Atlantic on a good day.
Nuclear early-warning systems would be effectively neutered. Due to the uncertain circumstances they were designed for, intercontinental nuclear missiles don't rely on outside systems for anything at all. They navigate by dead reckoning. ICBM detection systems, on the other hand, are bolstered by satellite monitoring to detect launches ASAP and ready defenses. There are other detection systems, but satellites give the earliest warning. With the loss of those satellites, everyone would have much less time defend against a nuclear ICBM and it'd be an ideal opportunity for one nation to strike another, or even these aliens launching one of these missiles in order to sow discord among the world powers and prevent them from cooperating.
How could they adapt? Not many options here. Observer posts all over the world near (known) launch sites, equipped with long-range radios would be the only option I can think of and it's a real stretch. This could well be the worst consequence of losing satellites.