I have a synology NAS and I don't understand the point of RAID 1 system. Why bother having a mirror? If I delete a file by accident it's deleted on both drives.
RAID is not a backup mechanism; it's a redundancy mechanism, and it does a completely different job – one protects against disk failures, the other protects individual files. So you wouldn't use one instead of the other; you'd normally use RAID along with backups or snapshots.
The main advantage of a redundant system is that it will not go down completely when a complete disk failure happens – the mirror allows you to continue using the NAS without interruption while the array is rebuilding.
(In other words: If you had a backup system but no RAID, you'd have to spend time restoring the complete system from backups every time a disk failed. If you have a backup system and a RAID 1 mirror, you only have to use backups when both disks fail at once, which happens much less often.)
Likewise, the redundant array should also allow you to replace disks that are only about to fail (e.g. if you see bad sectors increasing), and even to swap them with larger ones (if you're running out of space), without any downtime.
1 Doesn't apply to RAID 0.
Raid 1 isn't meant to protect you from deleted files. It only provides protection (redundancy) in the case of disk failure, wherein if one drive fails, the other has a complete copy of all your data.
RAID 1 consists of an exact copy (or mirror) of a set of data on two or more disks; a classic RAID 1 mirrored pair contains two disks.
This layout is useful when read performance or reliability is more important than write performance or the resulting data storage capacity.
The array will continue to operate so long as at least one member drive is operational.
Incremental backups, on the other hand, will help you in case you end up deleting a file by mistake, but if your main drive fails, you will lose all the data that hasn't been backed up.
It is always recommended that you use one in conjunction with another, such as by having your NAS work in a RAID 1 configuration, and then taking regular backups on a third drive (or possibly another RAID config!).
Some points the other answers have glossed over.
A backup is a point-in-time copy of your data. A RAID1 or higher array is a right-now redundant storage of your data. So if you did a daily backup (and it completed quick enough) then you could be restoring data that is up to 24 hours out of date. Can your use-case cope with losing a day's changes?
Cost - you mentioned in a comment that RAID1 feels wasteful. It is. But if the cost of losing your data is high enough then the cost of doubling the drives is miniscule. The cost of downtime also has to be considered.
Would I RAID and backup my /photos directory? Absolutely!
Would I RAID and backup my /TV+Movies directory? No, not at all.
If your budget is limited, RAID may not be feasable. However good backups are priceless. You can't replace some data like family photos, scans and documents.
TL:DR Backups are mandatory, RAID is optional.
If your aim is to protect against deletes, then RAID 1 is not for you.
RAID 1 will reduce the available disk-space by half, by making two disks serve as one disk, with the additional inconvenience that if one disk fails and is replaced, then the RAID might be inaccessible or very slow to access while it is rebuilding itself.
As your aim is backups, rather than sub-second data accuracy, you would be better served by using the two disks as two stand-alone disks and keeping two copies of the data, one perhaps somewhat behind the other.
With that simple setup, you would avoid the problems that can cause a RAID to fail, as some RAID failures may result in the total lose of data of both disks (some such cases are found on our site).
From your post and comments I get the impression that resiliency and resistance for wrong deletes are the most important to you. In that case two classic backups are better and safer than one RAID backup.
Aside from the redundancy and uptime benefits of RAID highlighted in other answers, there is another factor: data corruption.
A decent implementation of RAID will protect you from bit-rot (unrecoverable read errors) automatically. Backups only provide manual* protection against bit-rot. This is reduced if you discard old backups (you could lose the last intact copy of a file) or the backup medium itself suffers bit-rot.
You can further improve the catching of bit-rot by performing regular scans of your RAID, if the implementation allows it. Even better is to use a filesystem-based RAID like BTRFS or ZFS, where checksumming of data is done in software, reducing the reliance on disks to report ECC errors correctly.
If bitrot is something that concerns you, you should use RAID or a checksumming filesystem (plus backups). Ideally, use both.
* For example you could perform regular drive scans, and then cross-reference any read error sectors to files using filesystem debug tools, and then replace the referenced files with backed-up copies.
I have built about 20 RAID 1 systems as home computers for individuals who have no technical knowledge. Over the years, about half a dozen of these have had single drive failures. From the point of view of the non-technical user, nothing much happens when half of a RAID 1 fails - the computer continues to work, but gives an error message complaining about a problem with the RAID drive. The computer owner phones me, and I check out the problem. The solution is relatively simple - identify which drive has dropped out of the array, remove it, and install a new disk. The array is rebuilt over a period of perhaps 5 or 10 hours. The clients are impressed when I tell them that had I not insisted on building them a RAID system, they would have lost some or all of their files. But what about using a single drive with regular backups. The problem is that non-computer literate individuals struggle to successfully do a single backup (I have gone through coaching a senior citizen how to do it, and it isn't easy from their point of view). Forget about doing monthly backups, let alone backups every day.