The angel that appeared to Moses took the shape of a flame of fire.
"And the angel of the LORD appeared unto him in a flame of fire out of the midst of a bush: and he looked, and, behold, the bush burned with fire, and the bush [was] not consumed."  (Exodus 3:2)
Later, God used this angel as his official voice to speak to Moses.

The angel that appeared to Balaam was invisible to him, but very visible to Balaam's donkey.
"22  And God's anger was kindled because he went: and the angel of the LORD stood in the way for an adversary against him. Now he was riding upon his ass, and his two servants [were] with him.
23  And the ass saw the angel of the LORD standing in the way, and his sword drawn in his hand: and the ass turned aside out of the way, and went into the field: and Balaam smote the ass, to turn her into the way."
(Numbers 22:22-23)
Therefore, an angel may be visible or seen by one and at the same time invisible to others.   Later on the angel made himself visible to Balaam.
"Then the LORD opened the eyes of Balaam, and he saw the angel of the LORD standing in the way, and his sword drawn in his hand: and he bowed down his head, and fell flat on his face."  (Numbers 22:31)

When the Syrian army surrounded Elisha and his servant, Elisha saw the angels of God as  horses and chariots of fire, but his servant saw nothing.  After Elisha prayed, God opened the eyes of the servant, and he saw what was there all the time; the horses and chariots of fire."And he answered, Fear not: for they that be with us are more than they that be with them.  And Elisha prayed, and said, Lord, I pray thee, open his eyes, that he may see.  And the Lord opened the eyes of the young man; and he saw: and, behold, the mountain was full of horses and chariots of fire round about Elisha."  (2 Kings 6:16-17)

Another angel made himself visible to Peter, but invisible to the prison guards.
"And, behold, the angel of the Lord came upon [him], and a light shined in the prison: and he smote Peter on the side, and raised him up, saying, Arise up quickly. And his chains fell off from [his] hands."  (Acts 12:7)

There is the visible world where we move and have awareness through our senses, and there is at the same time the invisible world with its own activities and life that we cannot perceive.
"For by him were all things created, that are in heaven, and that are in earth, visible and invisible, whether they be thrones, or dominions, or principalities, or powers: all things were created by him, and for him:"  
Colossians 1:16
Angels can move from the invisible to the visible world or from the visible to the invisible world.  They can have objects such as swords, chariots and horses that can also be taken from the invisible to the visible world or go from the visible to the invisible.  The invisible world is just as real as the visible.

An angel appeared in a dream to Jacob and to Joseph.
"And the angel of God spake unto me in a dream, [saying], Jacob: And I said, Here [am] I."  
(Genesis 31:11)
"But while he thought on these things, behold, the angel of the Lord appeared unto him in a dream, saying, Joseph, thou son of David, fear not to take unto thee Mary thy wife: for that which is conceived in her is of the Holy Ghost."  (Matthew 1:20)

An angel appeared to Cornelius in a vision.
"He saw in a vision evidently about the ninth hour of the day an angel of God coming in to him, and saying unto him, Cornelius."  (Acts 10:3)

There are four events in the Bible where it states that the angels appeared as men.

 (1) The two angels that appeared to Lot, (2) the angel that appeared to Manoah and his wife,   (3) the angel that protected Shadrach, Meshach, and Abednego from the fire, and (4) the angels that were at the tomb of Jesus.  
1. "1  And there came two angels to Sodom at even; and Lot sat in the gate of Sodom: and Lot seeing [them] rose up to meet them; and he bowed himself with his face toward the ground;
2  And he said, Behold now, my lords, turn in, I pray you, into your servant's house, and tarry all night, and wash your feet, and ye shall rise up early, and go on your ways. And they said, Nay; but we will abide in the street all night................
5  And they called unto Lot, and said unto him, Where [are] the men which came in to thee this night? bring them out unto us, that we may know them."  (Genesis 191-2; 5)
2. "9  And God hearkened to the voice of Manoah; and the angel of God came again unto the woman as she sat in the field: but Manoah her husband [was] not with her.
10  And the woman made haste, and ran, and shewed her husband, and said unto him, Behold, the man hath appeared unto me, that came unto me the [other] day.
11  And Manoah arose, and went after his wife, and came to the man, and said unto him, [Art] thou the man that spakest unto the woman? And he said, I [am]."  (Judges 13:9-11)
3. "25 He answered and said, Lo, I see four men loose, walking in the midst of the fire, and they have no hurt; and the form of the fourth is like the Son of God.........
28 Then Nebuchadnezzar spake, and said, Blessed be the God of Shadrach, Meshach, and Abednego, who hath sent his angel, and delivered his servants that trusted in him, and have changed the king s word, and yielded their bodies, that they might not serve nor  worship any god, except their own God."  (Daniel 3:25; 28)
4. "And when they found not his body, they came, saying, that they had also seen a vision of angels, which said that he was alive."  (Luke 24:23)
"2  And they found the stone rolled away from the sepulchre.
3  And they entered in, and found not the body of the Lord Jesus.
4  And it came to pass, as they were much perplexed thereabout, behold, two men stood by them in shining garments:
5  And as they were afraid, and bowed down [their] faces to the earth, they said unto them, Why seek ye the living among the dead?"  (Luke 24:2-5)

The Hebrew word for angel is malak, which is always used in the masculine gender.