Strangely enough, one of the real problem areas I've run into in the mechanics of liqueur making is finding a good, general filtration system. Most of the fruits you'll use will break down a little over the course of steeping, and render themselves partly into a fine mist or paste. It's surprisingly hard to get this material out of the liqueur before bottling, and the consequences of not getting rid of it can be interesting. I've had my nectarine liqueur turn to jelly after a year or two, and I blame it on filtration.
The filtration problem isn't much of a problem if you just let all your "sludge" settle to the bottom of your jar, and siphon off the clear liquid from the top. This process is called "racking" in wine-making, and you'll find that wine recipes call for several rounds of racking your wine off the lees (things left behind). You wind up throwing away a small part of your batch, enough to hold the sludge at the bottom, but often it's a worthwhile tradeoff. The problem with racking your liqueur is the time involved, which can be quite long to get the solids to collect at the bottom. You could also just not squeeze your fruits (or squeeze them less than you otherwise could) when straining them, and have fewer solids to deal with that way. I've resorted to that occasionally with the nectarine. But for many flavors, I want to maximize my yield as best I can, and that means fighting with filters.
The Meilach book recommends initially starting out with coffee filters to remove remaining solids, and in the beginning I did that for a few months. But it requires amazing patience, and a lot of coffee filters. The particles of fruit block up the holes in the filters very quickly, so in short order the filters are completely clogged, and you have to put in another. It became clear to me that coffee filters were too fine a mesh to use for first-level filtration. Since those early days, I have started using paper towels as primary filters. They're much more porous than coffee filters, and consequently let a lot more solid material pass, but I hand-wave that problem away by reminding myself that the bottles will usually be consumed long before I have any problems.
The first indication that you have suspended solids in your bottled liqueurs is usually an unsightly ring about the meniscus (the little upward bend in the liquid level at the top of the bottle, where the liquid touches the glass). That ring around the inside of the neck is a result of solids which are lighter than the liqueur, and float to the top slowly. After a few months, they will have settled at the the top, around the outside of the liquid. And actually, that ring is usually nothing to worry about, only an aesthetic concern. But all the solids settling instead to the bottom, or remaining suspended in the middle, might cause troubles of a more serious nature, affecting the texture or flavor by their mere presence, or decomposing with time into other unwanted flavors.
Filtration usually comes in three distinct phases
The initial straining process can impact the later filtration stages to a certain extent. The finer a straining material you use, the less work your filters will have to do. I've found that a fine-mesh nylon bag can hold back a lot more macroscopic solids than cheese-cloth, probably because the cheese cloth weave stretches and deforms more under pressure. I expect that using woven cloth like linen, canvas or cotton would be a fine filtration, since it's a strong material and a tight weave, but I've never gotten around to using them myself. You may find that straining once under pressure (to squeeze the juice out of the fruit) and then again without pressure (pouring through a jelly bag, for instance), to pick up some of the particles that got through the first time, is worth the effort. Even a coarse-mesh nylon bag, used repeatedly, can easily remove all the large particles, so that you don't have to deal with them in the next pass.
The filtration-before-aging phase may not need to be as rigorous as the later filtration. I've found that a small amount of solids held in suspension during aging doesn't seem to harm the taste any. At this stage, I'll usually run through paper towels once (which may require a lot of paper towels, depending on the flavor), add the sugar, and seal the jar back up. There will doubtlessly be a noticeable cloud of solids in the jar, but since the jar will then be sitting still for a month or more, these solids will all fall to the bottom. This fact can help the final stage.
The post-aging filtration probably needs to be the most rigorous, unless you're lazy like me and don't care enough. Here is where the final consumers will see the result of your efforts, either in dense clouds at the bottom of the bottle, in rings at the top, or in the absence of these blemishes. Both are mighty unsightly, and anyone who drinks the dregs at the bottom will be unhappy.
So there are two things you can do here. One is to get a length of clear plastic hose (fish tank air hose works, as does tubing used in beer- or wine-making) and rack off (siphon) the unclouded liquid from the top; and the second is just filter through a series of finer filters until you've got all the solids out. I find that siphoning off the unclouded liquid first is useful if you've got a lot of heavy clouding (like with nectarines or pomegranates), where the amount of work to filter it out is daunting. It helps if you make large batches of the fruits that cloud like this, so you can ensure that you've got a good yield even if you have to throw away a lot of the clouded material.
But if you want to try to keep the extra liquid and separate it from the heavy clouding, then you'll need to run it through a battery of filters anyway. And if your clouding is light and distributed fairly evenly in the jar, then you have no choice. Until recently, the only two substances I'd found that were good for filtering were coffee filters and paper towels. These can be thought of as "fine" and "medium" filters. The "coarse" filters are things like jelly-bags and other nylon mesh bags. If thery're too coarse, multiple applications can help.
And that's actually an important lesson to remember. Passing the liqueur through the same grade of filter a few times is worth the effort. I've taken, when I have the time, to passing through paper towels twice with every batch, and I've found the results to be significantly better than just going once. And after passing through paper towels twice, the liqueur can go much faster through the coffee filters, allowing those filters to concentrate on the finer particles that the paper towels could never have caught.
Recently I started reading about wine-making, and discovered a whole world of industrial-strength positive-pressure filtration systems. The one I am most likely to acquire some day is the $85 Vinamat, a hand-pumped two-plate system, with a three-gallon capacity. These pump-filters use eight-inch-diameter filter pads which are about an eighth of an inch thick. They are as fine or finer than coffee filters, and work much faster. I'm not sure I will ever make the investment, though - I've found that I'm not so lazy that it's a problem with my liqueurs, and until I've got a good reason, in the form of an insoluble filtration problem, I'll probably never get around to it.
Be careful when using wine-specific filtration systems, though. They are designed to remove really small particles, and if you just dump liqueur which has only been through cheese cloth into it, you'll choke your pads very quickly. The pads run about three dollars a pair, and if you have to use more than one set per batch, then you're wasting money. So run the liqueur through coarse and medium filters a few times before sending them through your Vinamat or Polyrad. It'll be worth the effort.
On the other hand, don't use the absolute finest filters available for wine. Much of the flavor of some liqueurs is in extremely small suspended particles that aren't removed by coffee filters. There are wine filters fine enough to trap single-celled organisms, and those will completely denude your liqueur of its taste and body. Wine has to be extremely clear, but liqueurs don't. And often they aren't. Consider the difference between apple juice (as an analog to wine) and apple cider (as an analog to liqueurs).
One final note: early on, in my zeal to get all the juice from a batch of nectarine that I could, I mashed the fruit through a device my grandfather had given me, which is a conical collander with a wooden pestle designed specifically for mashing fruit. I rendered all the nectarines into a fine paste. Sadly, that fine paste suspended all the liquid inside, and I couldn't filter it out at all. Whenever I tried to apply pressure, all my filters either ruptured or began passing the solids right through, so I wound up losing most of the batch. I've never mashed my fruits again.
One good rule of thumb to remember, though, is, "Work with what you've got, and keep your eyes open." If you don't have a big wine-filtration system, see what you can do with paper towels and coffee filters. Pay attention to the sorts of results you get, and to how the liqueurs age. More often than not, you'll find that putting in a little effort in the early stages will yield all the results you could want, without having to go overboard.