For starters, we have a Hollywood/science fiction misconception as to how aliens would attack. War of the Worlds, etc. have basically given us the notion that aliens would buzz around shooting up civilians, blasting the bumbling military, and maybe taking a few prisoners for labor, food, whatever.
In my mind, this type of invasion plan just doesn't make sense (...yes, I know, I'm applying Human logic here...)
The hypothetical approach taken by alien invaders is going to be subject to a variety of factors, factors which will in turn affect the Earth's military/political response. They are as follows:
1. Culture/Mentalities. Perhaps the most important factors. What is the mindset and culture of the aliens? Do they value overwhelming force? Negotiation? Are they xenophobic or tolerant of other species and cultures? This is going to be the fundamental factor in determining how the aliens deal with the Earth situation. It's going to determine how they use diplomacy or if they even use it at all, and how they conduct their military campaign (whether they use WMD, target of civilian centers, etc.)
2. Objectives. Why are the aliens here in the first place and what are they after? Natural resources? Food? Slaves? A new colony? The answer to this question is going to determine what the aliens target and how much destruction they will be willing to do. Effectively, this factor determines the ROEs.
3. Resources. What kind of military, numerical, technological, and political resources do the aliens have on their hands? If the aliens have simply deployed a small expeditionary force, they might seek to take over a small foothold or bridgehead on Earth, rather than conquer the whole planet in one fell swoop.
4. Naval/Aerospace capability. (Superluminal drive technology, size of fleet). If the aliens are capable of moving at faster-than-light speeds, then Earth will have absolutely no advance warning. However, if they're moving at slower-than-light speeds, then obviously that changes things for both sides; for the Humans in that they have some warning, and for the aliens in that their travel time and ability to launch reinforcements are negatively affected.
This factor also leads to:
-4.1 Logistics: The aliens will be at the end of their rope conducting a long-range expeditionary attack. Their logistics will be severely taxed and this will probably limit their size of their force and the amount of ammunition they have, fuel they can expend, and the time they can operate independently before the force needs major resupply. In fact, they might even opt to establish a base of operations on Mars or the Moon before attacking Earth itself. This could take place over several years, allowing the aliens to stockpile enough ammo, food, and fuel to launch a prolonged campaign on Earth.
The way humans respond, is going to be reactive i.e. largely determined by what the aliens do first. However, a variety of Terrestrial factors will also intervene in dictating the Human response.
1. Geopolitical state of the Earth.If Iran suddenly acts up and the US deploys large amounts of air, ground, and naval assets to the Gulf, then that's obviously going to reduce the US military's ability to react to an invasion from space. Also, an international response will be key to fighting off an attack and the current level of global accord/dischord will regulate how effectively Earth nations will be able to unite and fight back.
India and China might jump at opportunity to show leadership in fighting the aliens, since it could allow them to increase their geopolitical clout. Post-invasion, assuming Earth beat off the attack, they would also stand to benefit done to other major power or superpowers.
2. Leadership. The initiative and level-headedness of military and political leadership at all levels will be key. At the bottom of the chain of command, infantry 2nd Lieutenants are going to have to quickly adapt to a dramatically different set of small-unit tactics in fighting alien ground forces. Meanwhile POTUS, the Russian PM, ChiCom Party leaders, and the Indian Prime Minister are going to need to show strong and co-operative leadership, and skill in diplomacy should negotiations arise.
3. Military focus. How Earth militaries are equipped in relation to the alien armed forces is going to prove decisive in how they chose to fight. For example, can F-22's evade the alien sensors? A yes or a no answer will lead to vastly different methods of employment of this asset. On a strategic level, the effectiveness of nuclear weapons or other WMDs versus alien units is going to dictate their use in campaigns.
Before we have answers to these questions, we can't begin to fill in enough of the matrix and determine exactly how the Earth and who would chose/be chosen to respond.
However, in my mind the major players would be:
United States of America
-Department of Defense
-relevant subordinate Joint Commands
-US Navy
-US Marine Corps (might be among the first military units to respond)
-US Army (would be a key player in a protracted land campaign
-US Air Force (Space assets and launch capability could prove key, along with its tactical and strategic airpower)
-Department of State (would prove important in an Human-alien diplomatic process)
-CIA/other intelligence agencies
China, India, Japan, the UN, Russia, the UK, etc. would also play a major role. The groups involved would probably be the opposite numbers to the US ones listed above.