Tips for Playing defense

It seems to me that I tend to make most of my mistakes while I'm down material or on the defense (in essence, a snowball effect).  Furthermore, when I'm doing puzzles, the ones I always mess up on are the ones that involve surviving attacks by your opponent.

Is this a common issue? and what strategies do you use to overcome it?

Yeah it's a common issue. Players tend to look at their own pieces more than their opponent's, so they will see threats they can make, but will miss what their opponent can threaten.

I think a good exercise is to solve puzzles in a particular way. First of all, don't make a single move until you think you have the whole solution. What you'll do is write down your intended solution to the puzzle... then you go over your solution slowly, one move at a time. Visualize each individual move, pause, identify the threat, and try to find ALL the ways the opponent can defend against it. If you find a way to defend you didn't consider, then your solution is bad, so start over... if you can't find any improvement for the opponent on any move, then your solution is probably good, and you can try it.

The point is you're going through the process of:
1) Clearly identifying what the threat is in your mind
2) Holding on to that position, and looking for multiple ways to defend

In a real game you'll be the defender, but it's the same idea... by looking for multiple ways to defend a single threat, you start to familiarize yourself with the types of moves that interfere with attacking ideas, which will make your defensive analysis more efficient.

An example of types of defensive moves...

Let's pretend a rook threatens a pawn on f2.

You can defend f2 directly.
You can move the pawn forwards or otherwise put a piece in between f2 and the rook.
You can threaten something equal to or greater than a pawn.
You can prepare to trap the rook on f2 after it captures.

So a good initial search is related to threatening something equal or greater. In almost every position it's useful to be aware of ALL the ways you can play a check or threaten the opponent's queen.

But anyway, this is in the beginner section so maybe that's a little advanced.

The most important thing is you slow yourself down, don't calculate a bunch of stuff like you normally would. Just look at the single position in front of you and identify what your opponent is threatening.

There are a few equally good options. Moving the king or queen pawn 2 squares forward on the first move are the most popular.

More important than the specific moves is that you're doing something called "completing development."

https://www.chess.com/article/view/the-principles-of-the-opening

@1

"while I'm down material"
++ while you are down material you generally lose. Do not defend then, but attack.

"or on the defense"
++ When you are on the defence, think carefully. Take time to think. Trade attackers if you can.

Here is a good defensive game.
White accepts new a gambit his opponent had studied for 8 years.
https://www.chessgames.com/perl/chessgame?gid=1095025 

Defense is tricky. In some ways, I think it's trickier than attacking -- attacks just seem more intuitive. A sloppy attack, and your opponent can still just buckle under the pressure and fail to find a good enough defense. A sloppy defense, and your opponent is likely going to break through.

Not only that, but it feels like defense has more important decisions. Should I defend by trying to launch a counterattack immediately, or should I defend by being passive and trying to hold my position? How do I arrange my pieces so that they can hold key squares, but they also won't run into each other? Should I try not to push my pawns, or should I make a pawn move to try to disrupt the enemy forces or to give my king space to run?

Here's a few ideas that I've found as I've played defenses in my games.

• The defender often wants to exchange off key attacking pieces. So, if my bishop that's defending can capture your knight that's attacking, I'll often want to make that capture. This tends to take the steam out of an attack, as the attacker will have to bring in reinforcements or else the attack will stop.
• The defender often has to make unintuitive or ugly moves. ...Nf8, for example, is sometimes a key defensive move to defend the g6 and h7 squares. Sometimes, black might have to play ...f6 to defend against a bishop targeting g7 (though black has to be very careful playing such a move!).
• The defender has to cover key squares. Even if my pawns start getting blasted off the board (by piece sacrifices, say), I still need to find ways to defend those key squares, usually by putting my pieces in such a way that they can cover those squares.
• The defender has to look for chances to counterattack. Often, attackers will have to leave one area of the board vulnerable in order to launch their attack. So, while the defender is covering key squares in their own camp, they should also be looking for ways to threaten key squares in the opponent's camp.
• The defender has to calculate more and be more precise than the attacker. The attacker can make a bunch of threats, but then decide not to follow through with them, and usually all that happens is that the attack stops and the attacker loses the initiative. The defender, on the other hand, doesn't have this luxury. The defender has to make sure that their defensive resources will hold. The attacker just has to calculate whether whatever attack they like most will succeed. The defender has to calculate whether any attack that the attacker decides on will succeed.

Reading through this bulleted list, you might ask yourself, well, why would anybody ever choose to defend? One answer is that sometimes you are just forced to defend, because of some mistake on the board earlier. But another answer is that a successful defense often leads to a productive counterattack. Psychologically, it might be more difficult for the attacker to change roles to the defender than it is for the defender to change roles to the attacker. Practically, the attacker may have sacrificed material, so any counterattack that the defender can launch, they'll be doing so with more material.

I know that some of this post is vague, but I hope that this at least gives you a few ideas on defending. If you're interested in learning more on defense, Andrew Soltis has a book New Art of Defense in Chess that I found to be a somewhat helpful primer on the topic. The games of Viktor Korchnoi are also known for their defense/counterattack themes -- Korchnoi himself had a positive lifetime score against Tal, a notorious attacker.

Here is another great defensive game to study
https://www.chessgames.com/perl/chessgame?gid=1042835