I have gathered that squishy essentially means "easily killed", but I'm not sure how to judge a champion's squishy-ness. What factors go into being squishy?
Squishiness is essentially how easily you fall during an engagement - the relative ease of which you can be squished. Whereas a tanky champion can be likened to the solidness of stone, a squishy one has the standing power of a pillow. It is one of the many factors of general survivability - the strict ability to withstand damage.
Generally, squishiness manifests in champions with low defensive stats, little or no defensive ability to prevent death under focus fire from the enemy team, and poor ability to trade blows. The general theme is, if you are squishy, your likelihood of making it through an ambush, or under fire in a team fight, is low to nonexistent.
You might notice that a lot of these are matters of "lacking". And that's what squishiness boils down to - lacking defenses. As it is easier to illustrate that way, example champions I'll list will generally be to show abilities that reduce squishiness - squishy champions thus being those who lack such abilities.
Low Defense Stats
Health, Armor, and Magic Resistance all play a vital role in not dying. Whereas a tank will excel in these stats (and further boost these with items), a squishy champion tends to have poor base stats and/or poor stat progression per level. Without even the weakest of defensive items, a squishy champion will fall to pieces when under attack.
Vladimir is an example of a champion with reduced squishiness in this department. His passive grants him extra AP for bonus health, and extra health for bonus AP. This encourages him to buy health items, while at the same time benefiting his health anyway while he builds offensively.
No Personal Death Prevention
Naturally, any champion, even a tank, will fall under enough focus fire, but squishies are those who die within moments of focus. This happens both from a lack of defense stats as before, but also from lacking abilities that help mitigate damage. This includes damage reduction, damage shields, or any method of instant damage negation - things that help once the damage actually starts to come in.
Poppy is an example of a champion with reduced squishiness in this department. Her passive sharply reduces damage, she gets stacks of Armor, and her ultimate lets her negate all damage except from one source. This both lets her survive once the damage starts, and even completely negate the effects of focus fire.
Poor Trading Blows
Likewise, being able to simply kill an enemy before they can hurt you doesn't affect your squishiness. That ability won't help you if you get jumped from the brush, or if you are on cooldown and took a wrong turn. If an initial burst doesn't take down the opponent, then a squishy hero may not survive the continued exchange of attacks with a foe, even if initially at a health advantage. Again, defensive stats play a large role in this, but also abilities to help you stick around in the fight without needing to retreat.
Graves is an example with reduced squishiness in this department. His passive increases his Armor and Magic Resistance, so he gets a growing advantage if he survives an ambush. As the rest of his kit also supports trading blows on the offensive end, his reduced squishiness really improves his survivability in that department.
Now that you see what factors (or rather, lack of factors) play a role in being known as “squishy”, what can you do about it?
The main things to look at are fixing your squishiness directly with items, or just mitigating its issues. Fixing it directly is basically - buy defensive items! Every little bit of health, armor, and magic resistance goes a long way if you’re a squishy champion. Many high tier items include those that give some measure of defenses in addition to offensive power (Abyssal Sceptre, Frozen Mallet, Wriggle’s Lantern). You may also consider items that directly present potent defenses - Banshee’s Veil, Hexdrinker, and Zhonya’s Hourglass all offer a variety of ways to handle your squishiness.
Mitigating it comes from equipment that helps your survivability otherwise. Lifesteal/spell vamp, for example, won’t save you from a nuke barrage, but it will improve your ability to trade blows by healing you for each hit (a fact which for AD champions is very nicely wrapped with Wriggle’s Lantern). Also consider items like Randuin’s Omen, which can slow down the enemy’s attack speed so that they cut through your buttery softness more like a dull knife than a laser beam.
A champion with a stun, silence, or snare can also use those abilities to stave off the flow of damage - if you don’t have one of your own, lane with someone who does. In fact, laning with someone who can protect you either with their own defensive shields or by holding off the enemy is a generally wise move.
Squishiness is a measure of how difficult a champion is to kill. It takes into account many different factors, and is best judged by simply playing and becoming familiar with all the champions. It's easier to judge through experience rather than trying to look at stats and comparing, although that can give you a head start.
That being said, squishiness is based on the champion's base health, health per level, base armor, armor per level, base magic resist, magic resist per level, and move speed. (This game sure has a lot of stats.) Some passives and active skills that champions have can also contribute. It can also be based on the items that are standard for that champion to get.
For example, Malphite has a lot of armor and his passive gives him a shield that essentially prevents real damage at the beginning of each skirmish. His Q allows him to escape, if necessary. He loves building armor items, since they interact so well with his skills, so he's generally really hard to kill.
Kog'Maw, on the other hand, has low move speed, armor, and health, and has no escape abilities at all. As a R-DPS character, his primary item focus should be on dealing damage, so he generally buys fewer health and armor items than his tankier counterparts. Even with a full item build, Kog'Maw is going to go down really fast compared to Malphite. He's squishy.
The factors that make a champion squishy or not are the base health, armor, and magic resistance and also the health gain, armor gain, and magic resistance gain per level.
In contrast to Beefy champions:
You also have to take into account abilities like health regeneration, shields, abilities that give extra armor or mr, etc.
You can check out base stats, as I did, from http://leagueoflegends.wikia.com/wiki/Base_champion_statistics
A squishy champion has:
Armor and Magic Resist reduce the amount of damage taken from Physical damage or Magical damage. The less the Armor and/ or Magic Resist a champion has, the more damage they will take, making them die faster. (making them squishier)
Hit Points tell you how much damage a champion can take before they die. The lower a champion's Hit Points, the quicker they will die. (the squishier they are)
Combine low Armor, low Magic Resist, and low Hit Points, and a champion is very squishy. On the flip side, that champion probably deals a lot of damage.
Every champion can be made to not be squishy. If you simply buy items that raise your Armor, Magic Resist, and Hit Points, your champion will survive longer, and be less squishy. This is a bad tactic.
Certain champions are supposed to deal lots of damage, and be squishy as a result. This is how they are supposed to function. If you neglect their damage, and make them tougher, they will have both mediocre damage, and mediocre defenses, rendering them nearly useless. The key is to find a comfortable balance, and that varies from champion to champion.
For guides on how to build champions, I suggest using http://www.mobafire.com/