It was very similar (at least superficially) with some notable differences.

The smallest coin would have been the penny as (sigh) now. The half-cent coin having stopped being minted in the 1850s, though I guess it could have been circulating still. There were coins in 1, 5, 10, 25, and 50 cent denominations. However, things aren't quite so simple as they would appear on the surface. Since, in principle, the value of coinage was partly due to the value of the metals they were made from. There were actually several different versions of 5 cent pieces, for example. There were 75/25 copper/nickel 5 cent pieces but there were also "half-dimes" made of mostly silver that were even smaller than dimes. Half-dimes stopped being minted in the 1870s but many would have still been in circulation.

There were also copper 2 cent pieces and copper/nickel plus silver 3 cent pieces (though, as above, the silver 3 cent piece wasn't minted any longer after the 1870s, but would still have been circulated).

The coins from the dime up to the dollar coin were silver. Here's where there starts being a greater difference. Gold coins were minted and circulated in 1, 2, 3, 5, 10, and 20 dollar values.

OK, here's where things get a little more varied. If you happen to have some US currency handy, go and take a look at it. At the top you'll see it says "Federal Reserve Note". Well, the astute reader may have realized that the Federal Reserve did not exist until 1913, so they very well didn't have such notes in the late 1800s, so what did they have? Unlike today, they didn't just have "one thing", they had several different things.

They had "Treasury Notes", which were a very clever dodge. Remember, at the time the money system of the US was based on metals, but the Treasury Note was not, it used one level of indirection by being based off coinage. In theory, if A=B and B=C then A=C, right? But in practice there were subtleties. You could, in principle, take a Treasury Note and have it exchanged for an equal denomination of coinage (e.g. take in one $100 note, get 5 $20 coins or 100 $1 coins). The trick here is that these were basically promissory notes that the Treasury would be obligated to redeem in coinage, however, they could choose to do so in either gold or silver coinage, or whatever mixture suited them. These were introduced and issued as silver mining started taking off and people started selling large amounts of silver bullion to the Treasury, but, of course, they were in general circulation as those folks spent them. In 1890/'91 the series of these notes printed by the Treasury came in denominations of 1, 2, 5, 10, 20, 50 ('91), 100, and a whopping 1000 dollars. For reference, $1k in 1890 would be the equivalent of nearly $30k today, so that's quite the compact store of value for its time (especially compared to 3.5+ lbs of gold coins or nearly 60 lbs of silver dollars).

There were also silver certificates, which were issued up through 1964, in fact. There's a whole complexity of the history of bimetallism and the gold standard in the US in this period which could be an entire separate thread which I'll skip. I'll just mention that these certificates started to be issued in 1878, and they were exchangeable for silver coinage. The initial runs were focused on higher values, with denominations of 10, 20, 50, 100, 500, and 1000 dollars, in 1886 1, 2, 5, 10, and 20 denomination certificates were issued. In 1890, for example, all of those denominations would have been in circulation, though in general silver certificates were rarer than other forms of paper currency.

And now, here's where things get well and truly messy. And, also, this is a yawning chasm of possibility that could be spun off into multiple other threads as well so I'll try to keep it succinct.

In addition to all of the above there was a truly astounding diversity of paper currency in the US, prior to the advent of the Federal Reserve it was somewhat of a decentralized system in many regards.

Let's start with "national banks". The name is slightly misleading as this was a whole ecosystem of over a thousand regional banks, each of which issued their own currency. Being a "national bank" meant that they fell under specific rules and regulations, especially in regards to reserves. Additionally, these banks were required to exchange notes from all other national banks at face value. So if you had a note for 5 bucks from the "National Bank of Jerkwater" you could still have some faith that it had some value since you could walk into any other national bank and exchange it for their notes. Additionally, while all of these various banks had their own designs, all of the notes were printed at the federal level to ensure consistency.

By the late 1800s the national banks had been substantially displacing the almost, but not entirely, unregulated "state banks". These were banks that were regulated at the state level (barely) and each issued their own currency. In principle these notes were also backed by precious metal coinage (specie), but in practice often the reserve requirements were played fast and loose, and if the bank went bust you were out of luck. The practical value of a given note from a given state bank was typically lower than its face value.

Not content to let the state banks have all the fun during the civil war the Union began issuing its own unbacked fiat currency, first as "Demand Notes" (colloquially known as "greenbacks"). The US government paid many of its debts by simply printing these out, but they fairly quickly became debased in value. To "fix" the problem the US then switched to having the US treasury print "United States Notes" which were legal tender (as coinage was). This was controversial, but it helped stabilize the value of greenbacks substantially, since they could be used to pay for "any debt public or private". In the 1870s congress passed a law which intended to retire most of the greenbacks in circulation by allowing them to be exchanged for coins, however, in practice most greenbacks remained in circulation. These United States Notes were issued in 1, 2, 5, 10, 20, 50, 100, 500, 1000, and 10k denominations. Unlike modern dollars they did not have a consistent design scheme from year to year of issuance. For example, the $1 greenbacks had Salmon P. Chase on the first versions printed in 1862, then George Washington afterward while the $2 notes had either Alexander Hamilton or Thomas Jefferson on them, while Andrew Jackson was on the $5 note in 1880 and the $10k note in 1878. In addition to Salmon P. Chase, there were several other faces on US notes that would be unfamiliar in relation to modern dollars: Daniel Webster, Lewis & Clark, Henry Clay, Albert Gallatin, John Quincy Adams, Joseph K. Mansfield, Robert Morris, DeWitt Clinton, and James Madison.

Additionally, it might be surprising, but many people in the 1890s would have kept their money in checking accounts (at either national or state banks). Though, of course, they would have used various forms of currency on a daily basis.

As for the value of money (or the cost of things), that depends on context, of course. A farm worker might make 10-20 dollars a month in the 1890s, whereas a railroad conductor might make $3 a day. Someone renting a room in a Boston tenement might pay around a dollar a week, while renting a multi-bedroom flat or entire house in San Francisco might have cost around $20-50/month.