In general, a "proxy" is a substitute or go-between. For example, a proxy server takes web page requests from a computer and transmits in on their behalf, and then passes the resulting page back. Using seven of them at once yields a similar level of defense as sitting in the middle of the Iron Cage.
In wargames, "proxy" is used as a verb: to "proxy" a model means to use some other model to represent it. For example, Jokaero look rather silly, but AT-43 Karmans look way cool, and they're made to the same scale, so people can and do use them as substitutes. Or, for another, Dark Eldar Mandrakes have awesome models, but crappy rules (in 5e this has been fixed), so people like to use their models to represent other, better units.
In tabletop terms there are a variety of different approaches to proxying stuff but they fall into three major schools of thought (presented in glorious technicolour and descending order of quality/respectability):
This is the practice of using existing models and making modifications to create either a personalised version of the original, or a different vehicle entirely. In either case the reason is normally to make better looking minis than are officially offered. Very minor conversions are extremely common, as practically all players like to customise their dudes and tanks to some degree and most modern plastic kits are built to be flexible and interchangeable so that minor conversions are as simple as gluing on different bits. More aggressive conversions can involve jamming together bits from multiple kits (known as kitbashing) or using practically no official parts of models at all (known as scratch building). Conversions that build from a single kit are practically always acceptable, and kitbashed or scratchbuilt models are normally fine as long as it's clear what they are supposed to be, what things they have on them and they fit the aesthetic. Various companies make conversion kits and parts to work with/around 40k models and specifically to fill in the gaps in terms of weapon options which are either never officially offered or only available from forge world. Of course for Orks you can pretty much please yourself.
This is fairly simple approach where you buy models (either from the same game or another) that look better or are cheaper and use them to represent something else. The key to finding acceptance in this school is in finding models that actually work in context. Being in the right scale is critical, and matching the aesthetic of the army is fairly important, you may get away with a Gundam model in a Tau army, but Space Marines? Not so much. Using single minis or a small group of them to represent specific things (characters or veteran units) is generally ok, they often look better or are way, way, WAY cheaper, and in friendly/unofficial games NOBODY is going to give a single fuck if you're not using Cadians, Catachan or the like to represent your Imperial Guardsmen (Seriously, 30$ for a box of ten Guardsmen).
This is normally the go-to for young or poor (or cheap) gamers playing friendly games, and even for veteran grown ups to test of how expensive things play without having to buy the units in question. Obviously these are deeply uncool, but at the same time if its what you have to do to play a game then beggars can't be choosers. Certainly 'official' games can't be played in this manner, but then again if you are doing this you almost certainly don't need to worry about playing those games. It's certainly not unheard of for big games like Apocalypse to be played with 95% of the forces as some kind of proxy simply because people want to play the game but it'll take thousands of pounds and months of diligent modeling and painting to make the forces most of which won't ever see the table again. Whole games can be played with proxies in this fashion (this gamer suggests using paper instead of random objects so you can write on them so you can't lose track of stuff) because we all get bored and since you're a gamer going outside or meeting attractive women is pretty unrealistic so you do what you gotta do. The absolute master of the random object stand in is the paper warhound titan, plans of which can be found on google. Print it out, cut along the dots, fold and tuck in the flaps and boom; a to scale titan.
The degree to which this practice is accepted officially depends on the company and setting. Battlefront Miniatures (makers of Flames of War) is pretty chill about proxies, while Games Workshop forbids them from their official tournaments and stores.
The skill of the conversion can make or break the acceptance: a Land Raider masterfully built out of Lego bricks (or a whole army, like this one) will likely be accepted (and maybe even complimented), while a Land Raider represented by a juice box probably won't.
One of the most common things to do is to proxy Drop pods with soda cans, they're about the same size, don't ever move, functionally useless once they hit the board, and most hobby stores at least sell soda, so they're readily available for younger players.
Trading Card Games
In many casual groups for TCGs (most notably Magic: The Gathering), a proxy card is a card that counts as another card. There's a couple of situations where this is relevant to the way people play.
- For deck testing, proxies are a fact of life. A group of people all need to be able to test any card available but actually getting hold of them can be extremely hard especially when a new set is released. When you stack that up with decks that they need to test against but won't be playing, the costs just keep going up. So proxies are what gets used a lot of the time to make sure that you can cover all the bases quickly and easily.
- There are tournaments (especially in extremely old formats where cards can be hilarious expensive, upwards of thousands of dollars) where proxies are in fact allowed. While these tend not to be official WoTC ones, they are still big, well organised events with very large prize pools. In these cases they generally let you proxy a set number of cards to try and get more people playing. That generally works because people always want to play Magic and especially the furiously complex formats where they can demonstrate their mental acuity (or lack of it) but can't spring the ten grand for a deck.
- In casual formats proxies are occasionally allowable. Many groups who play commander allow for a set number of proxies, either to keep the circle fresh and let people try things or to keep costs down in what can be an expensive format too. Similarly when people put together Cube sets they might use proxies too. Depending on how and why they do it, this is either very cool or very not cool. If it's because you won't shell out the eight bucks for a rare then that's not cool. When you do it to add 'power' to the cube without adding an extra few grand to the price tag, that's normally fine.
- Play with cards that don't actually exist. Always a casual event, except when official RND teams do this in private.
A side case for this are the really sexy, well produced proxies that some people have come out with. They are often as nice as official cards, although you can tell the difference. If you proxy using these kind of things then they become a LOT more acceptable. A borderless Jace, full art Dual Lands or any card with its modern text is almost always acceptable outside of the sanctioned formats. People sneer at proxies because they detract from the feel of the game. If a proxy looks sexy and conveys the right feel of power and awesome, as opposed to "feeling like a proxy", then no-one complains.
Additionally proxies help people be freer with who they play their cube with. These cards cost a LOT of money. Like two months salary lots of money. But they are small and very easily lost, damaged or stolen. When you play with people you trust then it's not that big of a deal, but you need eight to draft and sometimes that means you end up with friends of friends, or just that guy you met at a GP. Having a set of proxies for most/all of the expensive cards means you don't need to worry about that. The expensive stuff stays at home, while everyone still gets the same kind of fun from the experience. We're not screwing around when we say these things are expensive. A cube is something like 700 cards. Many of them will cost a few bucks, some will cost tens and some may cost hundreds. This is a stack of cards that will cost over a thousand dollars at a minimum, and often closer to five thousand, potentially up to tens of thousands. I hate to keep harping on it, but it's true. Proxies let you protect your investment from dicks and sticky fingers, and people understand that.
- Soda cans/bottles: The infamous carbonated Drop Pod. New or cheap 40k players often use these, sometimes two or three at a time. Anyone who even thinks of using them in tournaments will have multiple simultaneous and devastating offensive boots shoved up their buttocks.
- Toy tanks: Any suitably scaled Tiger II or Sherman can and has been proxied for any other tank in any other game. Notably, Leman Russes and Land Raiders are the most egregious proxy offenders both due to GW's outrageous prices, and lazy design, (hold a Land Raider model and a Mark V side by side some time)
- Game non-specific models: Some people try to cash in on other peoples' need for cheap models. These often look neat, and tend to hold enough respect that their owners can get away with only having about two tanks. And using them in every game. For every system. ALWAYS.
- Counts As, where an old unit is used as a proxy for a new one